RECENTADVANCES IN STOCHASTIC MODELING AND DATA ANALYSIS
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RECENTADVANCES IN STOCHASTIC MODELING AND DATA ANALYSIS 29 May - 1 June 2007
Chania, Greece
editor
Christos H Skiadas Technical University of Crete, Greece
r pWorld Scientific N E W JERSEY
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LONDON
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SINGAPORE
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BElJlNG
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SHANGHAI
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HONG K O N G
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TAIPEI
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CHENNAI
Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ojjice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
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RECENT ADVANCES IN STOCHASTIC MODELING AND DATA ANALYSIS Copyright 0 2007 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book, or parts thereox may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN-13 978-981-270-968-4 ISBN-10 981-270-968-1
Printed in Singapore by World Scientific Printers (S) Pte Ltd
This volume contains a part of the invited and contributed papers which were accepted and presented at the 12nd International Conference on Applied Stochastic Models and Data Analysis in Chania, Crete, Greece, May 29- June 1, 2007. Since 1981, ASMDA aims to serve as the interface between Stochastic Modeling and Data Analysis and their real life applications particularly in Business, Finance and Insurance, Management, Production and Reliability, Biology and Medicine. Our main objective is to include papers both theoretical and practical, presenting new results having potential for solving real-life problems. Another important objective is to present new methods for solving these problems by analyzing the relevant data. Also, the use of recent advances in different fields will be promoted such as for example, new optimization and statistical methods, data warehouse, data mining and knowledge systems and neural computing. This volume contains papers on various important topics: Stochastic Processes and Models, Distributions, Insurance, Stochastic Modelling for Healthcare Management, Markov and Semi Markov models, Parametric/ Non -Parametric, Dynamical Systems / Forecasting, Modeling and Chaotic Modeling, Sampling and Optimization problems, Data Mining, Clustering and Classification, Applications of Data Analysis and various other applications. The World Scientific had also published the proceedings in two volumes of the 1993 Sixth ASMDA Conference, held also in Chania, Crete, Greece. I acknowledge the valuable support of the Mediterranean Agronomic Institute, Chania, Greece, as well as the IBM France. Sincere thanks must be recorded to those whose contributions have been essential to create the Conference and the Proceedings. Finally, I would like to thank Anthi Katsirikou, Mary Karadima, John Dimotikallis and George Matalliotakis for their valuable support. Chania, July 30,2007 Christos H. Skiadas Editor V
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Contents Preface
V
1 Stochastic Processes and Models
1
An Approach to Stochastic Process using Quasi-Arithmetic Means Etienne Cuvelier and Monique Noirhomme-Fraiture The Quantum Generator of Translations in a Fraction-Dimensional Manifold Paulius MiSkinis Cause of Stock Return Stochastic Volatility: Query by Way of Stochastic Calculus Juho Kanniainen On a Class of Backward Stochastic Differential Equations and Applications to the Stochastic Resonance Romeo Negrea
2 10
18
26
2 Distributions
34
An Application of the Extended Waring Distribution to Model Count Data Variables Jose' Rodriguez Avi, Antonio Conde Sanchez, Antonio Jose' Sa'ez-Castillo and Ma Jose' Olmo Jime'nez Estimation of Simple Characteristics of Samples from Skewed and Heavy-Tailed Distributions Zdengk Fabian Estimating Robustness and Parameter Distribution in Compartmental Models of Neurons Noam Peled and Alon Korngreen On the Stability of Queues with Negative Arrivals Kernane Tewfk Random Multivariate Multimodal Distributions George Kouvaras and George Kokolakis A System Maintained by Imperfect and Perfect Repairs under Phase-Type Distributions Delia Montoro-Gazorla, Rafael Pe'rez-Ocdn and M. Carmen Segovia Asymptotically Robust Algorithms for Detection and Recognition of Signals Veniamin A. Bogdanovich and Aleksey G. Vostretsov
35
vii
43
51
59 68 76
82
viii
Recent Advances in Stochastic Modeling and Data Analysis
Three Parameter Estimation of the Weibull Distribution by Order Statistics Vaida Bartkute and Leonidas Sakalauskas
91
3 Insurance
101
Stochastic Models for Claims Reserving in Insurance Business Tarna's Falukozy, Ildikd Ibolya Vite'z and Miklds Aratd Stochastic Risk Capital Model for Insurance Company Gaida Pettere Measuring Demographic Uncertainty via Actuarial Indexes Mariarosaria Coppola, Ernilia Di Lorenzo, Albina Orlando and Marilena Sibillo Location as Risk Factor. Spatial Analysis of an Insurance Data-Set Ildikd Vite'z A Hierarchical Bayesian Model to Predict Belatedly Reported Claims in Insurances J&os Gyarrnati-Szabd and Lhszld Ma'rkus
102
4 Stochastic Modeling for Healthcare Management
145
Non-Homogeneous Markov Models for Performance Monitoring of Healthcare Sally McClean, Lalit Garg, Brian Meenan and Peter Millard Patient Activity in Hospital using Discrete Conditional Phase-Type (DC-Ph) Models Adele H. Marshall, Louise Burns and Barry Shaw Identifying the Heterogeneity of Patients in an Accident and Emergency Department using a Bayesian Classification Model Louise Burns and Adele H. Marshall Modelling the Total Time Spent in an Accident and Emergency Department and the Associated Costs Barry Shaw and Adele H. Marshall
146
5 Markov and Semi Markov Models
181
Periodicity of the Perturbed Non-Homogeneous Markov System M. A. Syrneonakiand P.-C. G. Vassiliou On the Moments of the State Sizes of the Discrete Time Homogeneous Markov System with a Finite State Capacity G. Vasiliadis and G. Tsaklidis Copulas and Goodness of Fit Tests Pal Rakonczai and Andra's Zernple'ni Discrete Time Semi-Markov Models with Fuzzy State Space Aleka A. Papadopoulou and George M. Tsaklidis An Application of the Theory of Semi-Markov Processes in Simulation Sonia Malefaki and George Iliopoulos
182
114 122
130 137
154
162
172
190
198 206 213
Contents ix
On a Numerical Approximation Method of Evaluating the Interval Transition Probabilities of Semi-Markov Models Dimitrios Bitziadis, George Tsaklidis and Aleka Papadopoulou Markov Property of the Solution of the Generalized Stochastic Equations Khaldi Khaled Partially Markov Models and Unsupervised Segmentation of Semi-Markov Chains Hidden with Long Dependence Noise Je'r6me Lapuyade-Lahorgue and Wojciech Pieczynski
22 1
6 Parametricmon-Parametric
242
x;
Independent Distributed in the Limit Components of Some Chi-Squared Tests Vassilly Voinov, Mikhail Nikulin and Natalie Pya Parametric Conditional Mean and Variance Testing with Censored Data Wenceslao Gonzcilez Manteiga, Ce'dric Heuchenne and Ce'sar Sknchez Seller0 Synthetic Data Based Nonparametric Testing of Parametric Mean-Regression Models with Censored Data Olivier Lopez and Valentin Patilea
229
234
243
25 1
259
7 Dynamical Systems/Forecasting
267
Application of the Single Index Model for Forecasting of the Inland Conveyances Eugene Kopytov and Diana Santalova Development and Application of Mathematical Models for Internet Access Technology Substitution Apostolos N. Giovariis and Christos H. Skiadas Exploring and Simulating Chaotic Advection: A Difference Equations Approach Christos H. Skiadas
268
277 287
8 Modeling and Stochastic Modeling
295
Likelihood Ratio Tests and Applications in 2D Lognormal Diffusions Ramdn Gutie'rrez, Concepcidn Roldan, Ramdn Gutitrrez-Sanchez and Jost Miguel Angulo Cartographical Modeling as a Statistical Method for Monitoring of a Spatial Behaviour of Population Irina Pribytkova Learning and Inference in Switching Conditionally Heteroscedastic Factor Models using Variational Methods Mohamed Saidane and Christian Lavergne
296
304
312
x
Recent Advances in Stochastic Modeling and Data Analysis
Correlation Tests Based NARMAX Data Smoother Validation Li Feng Zhang, Quail Min Zhu and Ashley Lorigden Kernel Based Confidence Intervals for Survival Function Estimation Dimitrios I. Bagkavos, Aglaia Kalamatianou and Dimitrios Ioannides Chaotic Data Analysis and Hybrid Modeling for Biomedical Applications Wlodzimierz Klonowski, Robert Stepien, Marek Darowski and Maciej Kozarski Stochastic Fractal Interpolation Function and its Applications to Fractal Analysis of Normal and Pathological Body Temperature Graphs by Children Anna Sods A Modeling Approach to Life Table Data Sets Christos H. Skiadas and Charilaos Skiadas An Extended Quadratic Health State Function and the Related Density Function for Life Table Data Charilaos Skiadas, George Matalliotakis and Christos H. Skiadas
322 330 338
342
350 360
9 Statistical Applications in Socioeconomic Problems
370
Dumping Influence on a Non Iterative Dynamics Ce'cile Hardouin Firm Turnover and Labor Productivity Growth in the Italian Mechanical Sector Luigi Grossi and Giorgio Gozzi Continuous Sampling Plan under an Acceptance Cost of Linear Form Nicolas Farmakis and Mavroudis Elefheriou A Dynamic Programming Model of a Machine Tool in Flexible Manufacturing Bernard F. Lamond Particle Filter-Based Real-Time Estimation and Prediction of Traffic Conditions Jacques Sau, Nour-Eddin El Faouzi, Anis Ben Aissa and Olivier de Mouzon Probability of Trend Prediction of Exchange Rate by ANFIS George S. Atsalakis, Christos H. Skiadas and Ilias Braimis The Organizational Structure of Greek Libraries: The State of the Art and the Perspective of Team Working Anthi Katsirikou
37 1 382
390 398 406
414 423
10 Sampling and Optimization Problems
433
Applicability of Importance Sampling to Coupled Molecular Reactions Werner Sandmann
434
Contents xi
Bispectrum Estimation for a Continuous-Time Stationary Process from a Random Sampling Karim Benhenni and Mustapha Rachdi Search via Probability Algorithm for Engineering Optimization Problems Nguyen Huu Thong and Tran Van Ha0 Solving the Capacitated Single Allocation Hub Location Problem using Genetic Algorithm Zorica StanimiroviC
442
11 Data Mining and Applications
472
Robust Refinement of Initial Prototypes for Partitioning-Based Clustering Algorithms Sami Ayramo, Tommi Karkkainen and Kirsi Majava Effects of Malingering in Self-Report Measures: A Scenario Analysis Approach Massirniliuno Pastore, Luigi Lombardi and Francesca Mereu The Effect of Agreement Expected by Chance on Some 2 x 2 Agreement Indices Teresa Rivas-Moya Qualitative Indicators of Libraries’ Services and Management of Resources: Methodologies of Analysis and Strategic Planning Aristeidis Meletiou
473
12 Clustering and Classification
511
Languages Similarity: Measuring and Testing Liviu P. Dinu and Denis Encichescu On Clustering Romance Languages Liviu P. Dinu and Denis Encichescu A Clustering Method Associated Pretopological Concepts and k-Means Algorithm T. V. Le, N. Kabachi and M. Lamure Alternatives to the Estimation of the Functional Multinomial Regression Model Manuel Escabias, Ana M. Aguilera and Mariano J. Valderrama A GARCH-Based Method for Clustering of Financial Time Series: International Stock Markets Evidence Jorge Cuiado and Nuno Crato
512
454
464
483
49 1
499
521
5 29 5 37
542
13 Applications of Data Analysis
552
Reliability Problems and Longevity Analysis Anatoli Michalski
553
xii
Recent Advances in Stochastic Modeling and Data Analysis
Statistical Analysis on Mobile Applications among City People: A Case of Bangkok, Thailand Pakavadi Sirirangsi Pollution Sources Detection via Principal Component Analysis and Rotation Marie Chavent, Herve' Gue'gan, Vanessa Kuentz, Brigitte Patouille and J e ' r h e Saracco Option Pricing and Generalized Statistics: Density Matrix Approach Petr Jizba
562 57 1
578
14 Miscellaneous
588
Inference for Alternating Time Series Ursula U. Miiller, Anton Schick and Wolfgang Wefelmeyer Estimation of the Moving-Average Operator in a Hilbert Space Ce'line Turbillon, Jean-Marie Marion and Besnik Pumo Monte Carlo Observer for a Stochastic Model of Bioreactors Marc Joannides, Irbne Larramendy-Valverde and Vivien Rossi Monte Carlo Studies of Optimal Stopping Domains for American Knock Out Options Robin Lundgren SONAR Image Denoising using a Bayesian Approach in the Wavelet Domain Sorin Moga and Alexandru Isar Performance Evaluation of a Tandem Queueing Network Smail Adjabi and Karima Lagha Assessment of Groundwater Quality Monitoring Network Based on Information Theory Malgorzata Kucharek and Wiktor Treichel Improving Type I1 Error Rates of Multiple Testing Procedures by Use of Auxiliary Variables. Application to Microarray Data Maela Kloareg and David Causeur
589
Author Index
597
605 613
62 1
630 636 645
653
CHAPTER 1 Stochastic Processes and Models
An approach t o Stochastic Process using Quasi-Arithmetic Means Etienne Cuvelier and Monique Noirhomme-Fraiture Institut d’Informatique (FUNDP) 21, rue Grandgagnage, 5000 Namur, Belgium (e-mail: e c d i n f 0 . f undp. ac .be, mnoQinf0 . f undp .ac .be)
Abstract. Probability distributions are central tools for probabilistic modeling in data mining. In functional data analysis (FDA) they are weakly studied in the general case. In this paper we discuss a probability distribution law for functional data considered as stochastic process. We define first a new kind of stationarity linked t o the Archimedean copulas, and then we build a probability distribution using jointly the Quasi-arithmetic means and the generators of Archimedean copulas. We also study some properties of this new mathematical tool. Keywords: Functional Data Analysis, Probability distributions, Stochastic Process, Quasi-Arithmetic Mean, Archimedean copulas.
1
Introduction
Probability distributions are central tools for probabilistic modeling in data mining. In functional data analysis , as functional random variable can be considered as stochastic process, the probability distribution have been studied largely, but with rather strong hypotheses , [Cox and Miller, 19651, [Gihman and Skorohod, 19741, [Bartlett, 19781 and [Stirzaker, 20051. Some processes are very famous like Markov process [Meyn and L, 19931. Such a process has the property that present is not influenced by all the past but only by the last visited state. A very particular case is the random walk, which has the property that one-step transitions are permitted only to the nearest neighboring states. Such local changes of state may be regarded as the analogue for discrete states of the phenomenon of continuous changes for continuous states. The limiting process is called the Wiener process or Brownian motion. The Wiener process is a diffusion process having the special property of independent increments. Some more general Markov chain with only local changes of state are permissible, gives also Markov limiting process for continuous time and continuous states. The density probability is solution of a special case of the Fokker-Planck diffusion equation. In preceding work [Cuvelier and Noirhomme-Fraiture, 20051 we used copulas to model the distribution of functional random variables at discrete cutting points. Here, using the separability concept, we can consider the continuous case as the limit of the discrete one. We will use quasi-arithmetic means in order to avoid copulas problem when considering the limit when the number 2
Approach to Stochastic Process using Quasi-Arithmetic Means 3
of cuttings tends to infinit,y. In section 2 we define the concept of distribution of functions and recall the notion of separability. In section 3 we propose to use the Quasi-arithmetic mean in conjunction with an Archimedean generator to build a probability distributions appropriate to the dimensional infiuite nature of the functional data. And in section 4 we study the properties of this new mathematical tool.
2
Distribution of a functional random variable
Let us recall some definitions that will be useful in the following paper.
Definition 1. Let (a,A, P ) a probability space and V a closed real interval. A functional random variable (frv) is any function from V x 0 + R such for any t E V ,X ( t , .) is a real random variable on (a, A, P ) . Each function X ( . , w ) is called a realization. In the following we will write for S ( . , w ) , and for X ( t , .). X,can be considered as a stochastic process.
x,
We study, here, the measurable and bounded functions.
Definition 2. Let 2, a closed real interval, then & ( D ) is the space of real measurable functions u(t) defined on a real interval V such that
-
Definition 3. Let f , g E C 2 ( D ) . The pointwise order between f and g on
V is defined as follows : &E
V >f ( t ) I y ( t )
f
I D 9
(2)
Definition 4. The functional cumulative distribution function (fcdf) of a f r v X on & ( D ) computed at u. E & ( D ) is given by : FX,D(U) = PIX
5 D
I.
(3)
Definition 5 . A frv is called separable if there exists in V an everywhere countable set I of points { t i } and a set N of f 2 of probability 0 such that for an arbitrary open set G c V and an arbitrary closed set F c JR the two sets { w : X ( t , w ) E F, Vt E
G)
{ w : X ( t ,w ) E F, Vt E G n I
}
differ from each other only on the subset N . The set 1 is called the separability set [Gihman and Skorohod, 19741. The space C 2(D ) is a separable Hilbert space. In the following we suppose that any realization of X is in Cz(V).
4 Recent Advances in Stochastic Modeling and Data Analysis
Definition 6. Two frv X , ( t , w ) and X z ( t . w ) (t E V , w E 0)are called stochastically equivalent if for any t E V P(X1(t,w)# X,(t.w))
=0
(4)
The interest of separability comes from the following theorem .
Theorem 1 (J.L. Doob). Let X and J’ be metric spaces, X be separable, y be compact. A n arbitrary random function X ( t , w ) , t E X with values in Y is stochastically equivalent to a certain separable random function.
3
The QAMM and QAMML distributions
In this section we build a sequence of sets that converge toward a separability set of V and at each step we define a probability distribution. Let n E N, and { t y , . . . t;}, n equidistant points of V such that t ; = inf(V) and t; = sup(D), and Vi E (1,. . . , n - 1) we have = A t . Let the two - tll = following sets
n n
A , ( ~ )=
{w 6 R : ~
( t p ,5 ~u(t:)} )
i=l
A(u) = { W E fl : i 5.0 U } We will use the following distribution to approximate the fcdf (3): P[An(u)] = H ( u ( t ; ) >... , ~ ( t : ) )
(5)
where H ( . , . . . , .) is a joint distribution of dimension 11. In previous works (see [Diday, 20021, [Vrac et al., 20011, [Cuvelier and Noirhomme-Fraiture, 20051) the Archimedean copulas were used for the approximation with small value of n. Let us recall the definition and property of copulas.
Definition 7. A copula is a multivariate cumulative distribution function defined on the n-dimensional unit cube [0, 11, such that every marginal distribution is uniform on the interval [0, 11 :
C : [O, 11,
+
[O, 11 ( ~ 1 ,. .. ) z L ~H) c ( ~ 1 ,; . .,
~ n )
The power of copulas comes from the following theorem (see [Nelsen, 19991).
Theorem 2 (Sklar’s theorem). Let H be an n-dimensional distribution function with margins F l , ..., F,. Then there exists an n-copula C such that for all 3: E R” , H ( x ~. ., . , ~ n = ) C ( F l ( x l ) ... ,
)
Fn(Xn)).
(6)
If F1, ...,F, are all continuous, then C is unique; otherwise, C is uniquely determined on Range of F1 x ... xRange of F,.
Approach to Stochastic Process using Quasi-Arithmetic Means 5 Before using copulas, we define a function that gives the distribution of the values of X,for a chosen t E V.
Definition 8. Let
a frv. We define the surface of distributions as follow :
G (tj Y) = PIX,5 Y1
(7)
We can use various methods for determining suitable G for a chosen value of t . Thus for example, if is a Gaussian process with mean value p ( t ) and standard deviation a ( t ) ,then we can use the cdf from N ( p ( t ) ,a ( t ) ) . In other cases we can use the empirical cumulative distribution function to estimate G:
In the following we will alway use this function G in conjunction with a function u of Cz (V): G [t,u ( t ) ] .So, for ease the notations, we will write G [t;u] = G [t,u ( t ) ] .If we use the preceding expression in conjunction with ( 6 ) , then (5) become :
P[A,(u)]= C ( G [ t ; ; u ]..., , G[tt;~])
(9)
An important class of stochastic process is the class of stationary processes. A stochastic process is said to be strictly stationary [Burril, 19721 if its distributions do not change with time; i.e. if for any t l ,..., t , E V and for any h E V ,the multivariate distribution function of . . , Xtn+h)does not depend on h. We propose here a more wide stationary property.
Definition 9. A stochastic process is said copula stationary if V t l , ..., t , E V and for any h E V ,the copula of ...,Xtnth)does not depend on h, i.e. its copula does not change with time. Let us notice that, if we deal with true functional data, realizations of a stochastic process X,we can suppose that there is always the same functional relation between X, and X t for any value s., t E V.If a frv is also a copula stationary stochastic process, then we call it a copula stationary frv. There is an important class of copulas which is well appropriate for copula stationary stochastic processes : the class of Archimedean copulas.
Definition 10. An Archimedean copula is a function from [0, 11, to [ 0 , 1 ]
where
4,called the generator, is a function from [0,1]to [0,ca]such that:
0
4 is a continuous strictly decreasing function,
0
d(0) = co and 4 ( 1 ) = 0,
Recent Advances in Stochastic Modeling and Data Analysis
6
Generator Uom. of 0
mame
Clayton to - 1 -1 - In ea--',+t , Frank , Gumbel-Hougaardl (- lnt)@ I -I
0
= 4-1 is completely monotonic on t in [0, m[ and for all k.
10,
0>0 B >o 0 2 1
m[ i.e.
( - ~ ) ' $ + ( t ) 2 0 for all
Notice that the k-dimension margins of (10) are all the same, and this for any value of 1 5 k 5 n. If X is a copula stationary f r v then expression (9) can be written :
PIAn(u)l =
+
L1
)
4 (G [tl;.I)
(11)
Table 1[Nelsen, 19991 shows three important Archimedean generators for copulas. The distribution (11) with the Clayton generator was already used for clustering of functional data coming from the symbolic data analysis framework (see [Vrac et al., 20011 and [Cuvelier and Noirhomme-Fraiture, 20051). Unfortunately the above limit is almost always null for Archimedean copulas when n + 0;) (see [Cuvelier and Noirhomme-Fkaiture, 2007])! Proposition 1. I f f o r u E &(D) : G ( t ;u ) < 1, Vt E
V ,then
Another objection t o the use of this type of joint distribution is something which we could call volumetric behavior. Definition 11. A function ZL E p , written Q p , if
&(D)is called a functional quantile of value
G(t;Qp) = p , Vt E 2)
(13)
The functional quantile Qp can be seen as the level curve of value p . Now let us remark that for a functional quantile :
P[An(Qp)]= lij
r
C 4(G [tl; i=l
Qpl)
1
= $ .(
' 4( P ) ) < P
+
(14)
And it is easy to see that, if n < m then $J [m#~ (p)] < [nq5(p)],and thus, the more we try to have a better approximation for a functional quantile of value p , the more we move away from reference value p toward zero. A simple way to avoid these two problems is to use the notion of quasi-arithmetic mean, concept which was studied by [Kolmogorov, 19301, [Nagumo, 19301 and [Aczel, 19661.
Approach to Stochastic Process using Quasi-Arithmetic Means
7
Definition 12. Let [ . , b ] be a closed real interval, and n E No. A quasiarithmetic mean is a function M : [a,b]" + [a,b] defined as follows:
where q5 is a continuous strictly monotonic real function. We show below that if we use the generator for Archimedean copulas in (15), we define a cumulative distribution function built from one-dimensional distributions. It's easy to prove the following lemma.
Lemma 1. Let n E No,F be a one dimensional cdf, and q5 a generator of Archimedean copula , then
is also a cdf. In various situations one can apply increasing transformations to the data without destroying the underlying dependence structure. This is classical in multivariate extreme value theory. And for these kind of transformation the copulas does not change. Proposition 2. Let n E No,{Fill5 i 5 ,n} be a set of one dimensional cdf, and q5 a generator of Archimedean copula , then
is a multivariate cdf. Proof. By the above lemma we have that the functions F ; ( z ) are cdf, and as q5 is an "Archimedean generator" so expression (10) is a copula, and thus 3 (Cy=lq5(F;(xi))) is a multivariate cdf. 0 We call the distributions given by the expression (17) the Quasi-Arithmetic Mean of Margins ( Q A M M ) distributions. Now if we use a Q A M M distribution in expression (5) :
then for each n E N we have an approximation, and the limit of the above expression is not always null.
8
Recent Advances in Stochastic Modeling and Data Analysis
Definition 13. Let : X be a frv, u E C z ( D ) ,G its Surface of Distributions and 4 a generator of Archimedean Copulas. We define the Quasi-Arithmetic M e a n of Margins L i m i t ( Q A M M L ) distribution of by :
In fact transformation (16) can be seen like giving an importance to the margins in proportion with the length of an interval [tl,tl+,l]in the approximation of Fxz,(u).
4
QAMML properties
First it is easy to see that the Q A M M L distribution preserves the functional quantiles.
Proposition 3. If I ; ~ C , D ( Q ~=) P
Qp
E
&(D) is a functional quantile of ualue p , t h e n
Now, what is the difference between the quasi-arithmetic mean of margins and the classical mean? Let
p
=
1
L
-
11~Il
G [t;u]d t
and let us define the function tp [t;u] = G [t;u] - p . Thus we can use the following Taylor’s approximation for all t (recall that 0 5 p , G [t;u] 5 1):
4 (G It;74) = 4 (P+ t p ( t I;). = 4(P)
+ 4’(P)tp It;I . + &”(P)-
t;
[t;u]
+ 0 (t; [t;4)
(21)
and then
and so like $ is a decreasing function, 4’’ ( p ) 2 0 and as u a r ( t ) 2 0 we can see that FX ~ ( udecreases ) with the variance of the differences between the function u-knd the quantile function associated to the value of the arithmetic mean of G along u.. And so the Q A M M L distribution is equal to the arithmetic mean only in the case of quantile functions.
Approach to Stochastic Process using Quasi-Arithmetic Means
9
Conclusion I n this paper we do n o t propose a new method in Functional Data Analysis but a new Probabilistic tool. Like in t h e real case, we can hope t h a t this tool can be used for analysis of functional data, like in mixture decomposition, statistical tests, ... Moreover, several ways to improve t h e tool exist. B y example let us note that the QAMML definition (see (19)) uses a uniform distribution over D: other distributions could be considered (see [De Finetti, 19311).
References [Aczel, 19661J Aczel. Lectures o n Functional Equations and Their Applications. Mathematics in Science and Engineering. Academic Press, New York and London, 1966. [Bartlett, 1978lM S Bartlett. An introduction to stochastic processes. Cambridge University Press, Cambridge, 1978. [Burril, 1972lC W Burril. Measure, integration and probability. McGraw-Hill, NewYork, 1972. [Cox and Miller, 1965lD R Cox and HD Miller. The theory of stochastic processes. Methuen, London, 1965. [Cuvelier and Noirhomme-Fraiture, 2005lE. Cuvelier and M. Noirhomme-Fraiture. Clayton copula and mixture decomposition. In ASMDA 2005, pages 699-708, 2005. [Cuvelier and Noirhomme-Fraiture, 2007lE. Cuvelier and M. Noirhomme-Fraiture. Classification de fonctions continues l’aide d’une distribution et d’une densit dfinies dans un espace de dimension infinie. In Extraction et gestion des connaissances EGC’2007, pages 679-690, 2007. [De Finetti, 1931lB De Finetti. Sul concetto di media. Giornale dell’ Instituto Itialiano degli Attuari, 2:369-396, 1931. [Diday, 2002lE. Diday. Mixture decomposition of distributions by copulas. In Classification, Clustering and Data Analysis, pages 297-310, 2002. [Gihman and Skorohod, 197411I Gihman and A V Skorohod. The theory ofstochastic process. Die grundleheren der mathematischen wissenschaften in einzeldarstellungen. Springer, Berlin, 1974. [Kolmogorov, 1930lA Kolmogorov. Sur la notion de moyenne. Rendiconti Accademia dei Lincei, 12(6):388-391, 1930. [Meyn and L, 19931s P Meyn and Tweedie R L. Markov chains and stochastic stability. Communications and Control. Springer-Verlag, New York, 1993. [Nagumo, 1930lM Nagumo. Uber eine klasse der mittelwerte. Japan Journal of Mathematics, 7:71-79, 1930. [Nelsen, 1999lR.B. Nelsen. An introduction to copulas. Springer, London, 1999. [Ramsay and Silverman, 200515 0 Ramsay and B W Silverman. Functional Data Analysis. Springer Series in Statistics. Springer, New-York, 2005. [Stirzaker, 2005lD Stirzaker. Stochastic processes and models. Oxford University Press, Oxford, 2005. [Vrac et al., 2001lMathieu Vrac, Edwin Diday, Alain Chkdin, and Philippe Naveau. Mklange de distributions de distributions, d6composition de melange de copules et application B la climatologie. In Actes du VIIIkme congre‘s de la Socie‘te‘ Francophone de Classijication, pages 348-355, 2001.
The quantum generator of translations in a fraction-dimensional manifold Paulius MiSkinis1>2 NORDITA, Nordic Institute of Theoretical Physics, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark Vilnius Gediminas Technical University Faculty of Fundamental Sciences Department of Physics Sauletekio Ave.11, LT-10223, Vilnius-40, Lithuania (e-mail: paulius .miskinismfm. vtu.It)
Abstract. In the case of the quantum generalization of random processes with the Hurst index H # 1/2, expression for the quantum Hermitian generator of translations and its eigenfunctions are proposed. The normalization constant has been determined and its relation to the operator of momentum is shown. The interrelation between the momentum and the wave number has been generalized for the processes with a non-integer dimensionality a. Keywords: quantum generator of translations, fractional derivative, non-Markovian process, Hermitian operator.
1
Introduction
The physical basis for the existence of quantum mechanics comprises a series of phenomena described by the mathematical theory of the Wienerian processes. The non-Markovian stochastic process is a natural generalization of the Brownian motion or the Wiener stochastic process [l]. The foundation for this generalization is the theory of stable probability distributions developed by Ldvy [2]. The most fundamental property of such distributions is the stability in respect to addition, in accordance with the generalized central limit theorem. Thus, from the probability theory point of view, the stable probability law is a generalization of the well-known Gaussian law. The nonMarkovian processes are characterized by the Hurst index H , which' takes values 0 < H < 1. At H = 1/2 we have the Gaussian process or the process of the Brownian motion. The non-Markovian stochastic process with stable Ldvy distributions is widely used to model a variety processes such as anomalous diffusion [3], turbulence [4], chaotic dynamics [5], plasma physics [6], financial dynamics [7], biology and physiology [8] (for recent references see e.g. [9-111). 10
The Quantum Generator of Translations 11
The constantly increasing number of experimental facts in various fields of knowledge related to classical non-Wienerian processes evokes a natural desire to “close” the commutative diagram shown in Fig. 1. QM
lh WP
a FQM
lh
a SLP
Fig. 1. Schematic representation of interrelations of Wienerian processes (WP), stable LBvy processes (SLP), quantum mechanics (QM) and fractional quantum mechanics (FQM). and, at least formally, to consider the possible existence of a quantum analogue of a more narrow class of phenomena related to stable Levy processes, the so-called fractional quantum mechanics (FQM) [12-141. Unfortunately, these works are not aimed at a thorough analysis of the properties of the quantum operator of momentum. The lack of such analysis results in some accuracies of even gross blunders while formulating FQM (see Conclusions). The present note offers a brief discussion of one of the crucial issues related to FQM, which is the one-dimensional operator of momentum. Like in usual quantum mechanics (QM), one-dimensional problems are a kind of excess idealization. Nevertheless, they may be used for elucidating the fundamental features of FQM. One-dimensional problems arise while considering the three-dimensional evolutionary equation in which the interaction potential depends on a single coordinate. This fact allows, with the aid of a corresponding factorization, to move to a simpler one-dimensional evolutionary equation. The purpose of this paper is formulation in the explicit form quantum expression of the one-dimensional operator of momentum for the fractional probability processes with a non-integer dimensionality a and the investigation of the interrelation between the quantum generator of translations and the operator of momentum.
2
Fractional quantum operator of momentum
The classical definition of momentum in QM follows from the invariance of the Hamiltonian of the quantum system H with respect to the infinitesimal displacements Sx [15]. Under such transformation, the wave function +(z) turns into the function
+(x here
+ b z ) = +(x) + C5xax+ = (1+Sxa,)
+(z),
axis the differentiation operator over the space variable x.
12 Recent Advances in Stochastic Modeling and Data Analysis
ax$
However, it may turn out that does not exist, but there exists the so-called fractional derivative a:$ in which the order of the derivative a may be both an integer and a fractional number. For the function determined on the whole real axis R, the right and left derivatives of the order a are derived
where [a]and { a } are the integer and the fractional parts of the parameter a. For the bilateral derivatives to exist, it is sufficient that $ ( x ) E C[a](52), where C[a](f2) is a set of continuously differentiated functions of the order [a]determined on the domain 52 [El. Another peculiarity related to the operator of momentum is the expansion of the wave-function $ ( x ) into a Taylor series by fractional powers [17,18]
c [a1
$(x) =
(x - xO)=+n + Rn(x)
7
(3)
n=O
2'
where are numerical coefficients and R,(x) is he residual term, which provides a better approach to the initial function. n all such cases, determination of the quantum operator of momentum should be specified. It is reasonable to suppose that the momentum operator should be proportional to the fractional derivative:
E
@ = ca;$(x)
(4)
here C is a certain coefficient of proportionality. For a -+ 0, we must obtain a usual quantum operator of momentum, 5 = Thus, in FMQ we always deal with two kinds of limit transitions: 1) ti -+ 0, when we shift to classical mechanics, and 2) a -+ 0, when we turn to usual QM (see Fig. 1). The kind of the coefficient C in the expression for the momentum (4) is best defined if on the whole real axis we consider a flat wave of the form
-%ax.
there fi is the Planck constant and 1, is a certain peculiar scale of the length of the nonlocal process under consideration. Let us impose a requirement for the momentum operator (4) to obey the eigenvalue equation p$ = p $ . Applying the property of the fractional derivatives, d;ienx = neenx (ReK > 0), we obtain that
c = (--i)QfiZga--l.
(6)
For the values observed in QM to be real, the corresponding operators should be Hermitian. It is easy to see that the quantum operator of momentum (4) with the constant C from (6) is non-Hermitian. In order to obtain
The Quantum Generator of Translations 13
a Hermitian operator of momentum, to the type (4) operator we will add a Hermite-conjugated operator $+ ; then, the momentum operator determined in this way
-
here is the symbol of transposition, will be clearly Hermitian. Indeed, the momentum operator (7) fi = ($+ + $ - ) / 2 will be Hermitian because of the idempotency of the operation of Hermitian conjugation ((@+)+= $) and the structure of the operator itself ($- = (@+)+).On the other hand, employing the rule of fractional integration by parts, +m
p* =
1,
'p*$+$dx = 2
/
+m
cp* (@-+@+)$d3: = P
-m
,
(8)
we directly see that the momentum operator is Hermitian for the different functions of state 'p and $. Thus, we obtain that the operator (7) is Hermitian and its eigenvalues on the whole real axis are flat waves of the (5) type. Like in the classical case, the eigenvalues of the momentum operator do not belong to the class L 2 ( R ) . Therefore, they don't describe the physically realizable states of the quantum particle. These eigenfunctions should be regarded as the basic functions, which comprise the complete system of functions.
3
Wave function normalization
To determine the constant normalization A in the expression for the flat wave ( 5 ) , we will take that
(9)
s
This is a particular case of the conventional condition $p,$pdx = 6(p' - p ) for p' E 0. Using the property of the &function, we shall obtain that
Let us specify the peculiarities of such normalization. Firstly, generally speaking, the amplitude is a complex magnitude; secondly, it depends on the eigenvalue of the momentum p . Only when a -+ 1, as the case should be,
A + l / m . Inasmuch as the physical sense applies not to the amplitude itself but to IAI2, the complex nature of A does not contradict unitarity. However, because of the complex nature of the amplitude we may get an impression that we deal with a damping wave; however, actually there is no damping, because A # A ( z , t ) . Besides, the same conclusion results from analysis of
14 Recent Advances in Stochastic Modeling and Data Analysis
the dispersion expression of the corresponding Hamiltonian. The dependence A = A(p)is not a matter of principle and may be avoided by a suitable choice of the normalization condition. For instance, under condition
dx = 6 ( ~-' K ) ,
$:t$&
(11)
the dependence A(p) is absent. Another important circumstance should be noted as regards the type of the momentum operator. Transition to momentum representation is not a Fourier transformation. Momentum representation should be understood in the sense of f-representation:
4
Translation operator
Lastly, let us derive the formula to express, through the momentum operator
6, the parallel translation operator in space to any finite (not only infinitesimal) distance. From the definition of such an operator it follows:
Fz+$(x) = $(x
-a),
F?-$(x)
= $(x
+ b) ,
(13)
In this case, a and b denote the values of finite displacements but not the coordinate ends of the interval. Expanding the function $(x - a ) in the neighbourhood of the point x into a Taylor series by fractional powers as in (3) and employing the expression for the "right-hand" and "left-hand" parts of the momentum operator,
$7 = - i h i ; - l q ,
6:
= irii;-l@,
(14)
we obtain that
where EE is the generalized exponential function. These are exactly the finite displacement operators we have been searching for. For a -+ 1, for Tz+ we obtain that
The expression T?- could be obtain by substituting in equation (16) a -+ -b and 6 : -+ p?.
The Quantum Generator of Translations 15
5
The shift operator in a superspace of fractional dimension
The modern theories of quantum field specifically unite physical fields of various statistics. Physical fields are unified into super-multiplets which are irreduceable 2,-graded Poincar6 group representation. 8, In the algebra of this group] alongside the translation generators P, in the ususal space, translation generators in the odd-dimensional sector the - i (70) 8, is present. generator Q = In this part of the paper we consider a generalization of the shift generator Q for the case of fractional dimensional of the odd-dimensional sector of the Minkowski superspace. In this case, differently from the shift generator in the even sector, we cannot rely upon physical intuition, because the odd Grassman variables and parameters are unobservable. To obtain the fractional shift operator Q, we shall employ the Balakrishnan formula, konown form functional analysis [191: N
&
By analogy with P,,
Q = lim E-' (T'
-
I ) , where TE= eEQ ,
E'O
(19)
is an element of Grassman's algebra. We take E" as a matrix operator for which, taking into account its Grassmanian nature,
E
( I + E)"
=I
+
a!&
(20)
is an exact equality. Threfore
For translation operators in the even sector, formula (21) leads to the already discussed operator of the fractional Marchaud derivative a;. Therefore, expression (21) may be regarded as the odd Marchaud derivative. Note here that, differently from the usual fractional Marchaud derivative] integration in (21) over E takes place in the whole space of spinor determination] but not within certain area. Formula (21) allows generalization for the c a e of a! > 1.
16 Recent Advances in Stochastic Modeling and Data Analysis
6
Conclusions
Note that the classical restriction on the smoothness of the wave function $(x) E C2([a,b ] ) does not hold here. The restriction on $(x) follows from the continuity equation; however, in the case of fractional dimension we can show that the condition of continuity is changed, and the limitation on $(x) is reduced to $(x) E d a ] ( [ ab, ] ) . Another note pertains to the structure of the momentum operator. It seems highly significant that the momentum operator consists of two parts the right and left displacements. In classical fractional mechanics, it is quite possible to limit ourselves to one of these two components - $+ or p-. In the quantum case it is impossible, because the full operator of momentum is a Hermitian. The limit transition h + 0 for { a }# 0 means transition to classical fractional mechanics. However, the form of the momentum operator undergoes no qualitative change: fi = $ (p+ fi-), i.e. it consists of two parts, each being proportional to its one-sided derivative. For linear evolutionary equations of classical (not quantum) fractional mechanics this type of structure of the momentum operator may be simplified if p = @+ or p = Ij-. However, here additional considerations are necessary. For the nonlinear fractional evolutionary processes it is impossible in principle, because fi+ = 6- is the condition of smoothness. Note here, too, that all results for the Hermitian operator of momentum are valid in the case of the Riesz quantum derivative: RD,"$(x) K [(-ia)T
+
+
(8) ]: $(x). From the definition of n there follows an interrelation between the momentum and the wave number:
Q: + 1, tc + Ic, and the expression (22) turns into p = hlc. The appearance of the characteristic length scale of I , and the power dependence of the quantum particle momentum on the wave number directly indicate the fractional character of quantum mechanics. Thus we have the Hermitian quantum operator of momentum (7) with the eigenfunctions (5). This allows us to construct the quadratic form of the Hamiltonian H K p 2 instead of the power form H K D,Jp la, and the Hermitian Hamiltonian instead of non-Hermitian proposed in [12,13],and the unitarian Hamiltonian instead of non-unitarian proposed in [14].
for
References [l]M. Kac. Probability and Related Topics in Physical Sciences. Chap. IV., Interscience, New York, 1959.
The Quantum Generator of Translations 17
[2] P. LBvy. The'orie de 17Addition des Variables Ale'atoires GauthierVillaws, Paris, 1937. [3] B.B. Mandelbrot, J.W. van Ness. Fractional Brownian Motionx, Fractional Noises and Applications. SIAM (SOC. Ind. Appl. Math.) Rev. 70(4): 422-437, 1968. [4] J. Klafter, A. Blumen and M.F. Shlesinger. Stochastic pathway to anomalous diffusion. Phys. Rev. A 55(7): 0081-9085, 1987. [5] G.M. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos. Physicu D 76(1-3): 110-372, 1994. [6] G. Zimbardo, P. Veltro, G. Basile and S. Principato. Anomalous d i f i sion and Lvy random walk of magnetic field lines in three dimensional turbulence. Phys. Plasmas 2(7): 2653-2163, 1995. [7] R.N. Mantega and H.E. Stanley. Scaling behaviour in the dynamics of an economic index. Nature 376(6535): 46-48, 1995. [8] B.J. West and W. Deering. Fractal physiology for physicists: Lvy statistics. Phys. Rep. 246(1-2): 1-100, 1994. [9] A. Le Mehautk et al. (Eds.) fiactional differentiotion and its applications. Books on Demand, Norderstedt, 2005. [lo] R. Metzler and J . Klafter. The restaurant at the end of the random walk: rkcent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37: R161-R208, 2004. [ll] G.M. Zaslavsky. Chaos, fractional kinetics, and anotaloub transport. Physics Reports 377: 461-580, 2002. [12] E. Lutz. Fractional Transport Equations for LBvy Stable Processes, Phys. Rev. Lett. 86(12): 2208-2211, 2001. [13] N. Laskin. Fractional quantum mechanics. Phys. Rev. E 62(3): 31353125, 2000. [14] M. Naber. Time fractional Schrodinger equation. J. Math. Phys. 45(8): 3739-3356, 2004. [15] H. Kleinert. Path Integrals in Quantum Mechanics, Statistics and Polymer Physics. World Scientific, Singapore, 1990. [16] V. Marchaud. Sur les dBrivBes et sur les differences des functions de variables rkelles. J. math. pures et uppl. 6(4): 238-235, 1927. [17] S.G. Samko, A.A. Kilbas and 0.1. Marichev. Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Amsterdam, 1993. [18] P. MiSkinis. Nonlinear and nonlocal integrable models. Vilnius, Technika, 2003. [19] E. Hille, R. Phillips. Functional analysis and semigroups. Cambr. Univ. Press. 1961.
Cause of Stock Return Stochastic Volatility: Query by Way of Stochastic Calculus Juho Kanniainen’ Tampere University of Technology Institute of Industrial Management P.O.Box 541, FI-33101 Tampere, Finland (e-mail: juho. kanniainen(0tut. f i) Abstract. This study uses stochastic calculus to investigate the causes of the stock return stochastic volatility. The study aims to advance new explanations of stochastic volatility that hold also if the firm is unleveraged, and if the level of uncertainty about future business conditions does not change. Using the dividend discount model, I show that stock return volatility is admittedly stochastic if future dividends are affected by more than one stochastic state variable. Morever, I study how the mappings of state variables are related to stochastic return volatility. This study also investigates the effects of the discount rate and state variables’ mutual correlation on the level of stock volatility and its fluctuation, finding substantial relationships therein. Keywords: Stochastic calculus, Stochastic volatility, Geometric Brownian motion,
Dividend discount model.
1
Introduction
It is by now a widely accepted observation that the volatilities of individual stocks and aggregate stock markets are not constants, but change stochastically over time. The literature has repeatedly presented sophisticated statistical models t o describe stochastic volatility, but the question of why stock return volatility varies remains open. [Schwert, 19891 presents an extensive analysis of the relation between market volatility and economic activity, confirming Officer’s [Officer, 19731 earlier results that market volatility is higher during economic downturns. This has also been justified by [Hamilton and Lin, 19961, who found that economic recessions are the single most important factor explaining market volatility, accounting for about 60 percent of its variation. Further, changes in volatility are also related t o financial and operating leverage, personal leverage, interest rates, inflation rates, money growth, industrial production growth, trading volume, trading halt, and program trading (see, for example, [Black, 19761, [Christie, 19821, [Mascaro and Meltzer, 19831, [Schwert, 19891, [Schwert, 19901). Stock volatility is also stochastic if a stock has option characteristics; that is, the stock can be viewed as an option on the leveraged firm’s assets (see [Merton, 1974]), and the firm may have numerous growth options. Overall in the literature, the usual explanation of stochastic volatility is either corporate leverage or a change in the level of uncertainty about future macroeconomic conditions. 18
Cause
0.f
Stock Return Stochastic Volatility 19
This study advances new explanations of stochastic volatility by way of stochastic calculus. My explanation does not challenge the earlier explanations but rather complements them. This paper investigates the significance of the above observation by suppressing the other possible causes of stochastic volatility. The assumption of risk-neutrality suppresses leverage effects, the assumption of a constant interest rules out the possibility that the randomness of stock volatility is driven by varying interest rates, and the assumption of constant state variable volatilities eliminates any change in the uncertainty about future macroeconomic conditions. I also examine the effects of state variables’ mutual correlations and the discount rate on the level of volatility and its variation.
2
Model
To suppress the leverage effect, we assume that the risk-neutrality, and hence the discount rate, denoted by p, equals the risk-free interest rate. Moreover, by assuming a constant interest, we rule out the possibility that the randomness of volatility is driven by varying interest rates. The stochasticity of stock volatility arise solely from the dividend process. We denote the dividend stream by {Dt,, D t z , . . . , Dt,}, where dividends occur at (known) times t l , t 2 , . . . ,t,. The stock price is assumed to equal the cumulative present value of its expected dividends, n
v(t)=
Et [Dt,]exP [ - p ( t k
-
t)],
(1)
k=l
Suppose that a discrete dividend stream consists of n discrete dividends. All investors are assumed to monitor the processes of state variables and continuously revising their beliefs regarding to expected dividends. Each dividend is driven by an m stochastic state variable. In addition, the dividends’ state variable vectors need not be equal; that is, the next year’s dividend can be driven by state variables different from those of the dividend paid after five years. Consequently, we specify the dividend process in rather general terms and allow that, for example, each dividend depends on the interest rate and inflation, but that the two first dividends also depend on the oil price, whereas later dividends depend on the price of biodiesel instead of oil. Overall, the matrix X ( t ) E Rmxn, represents the dividend stream information apprearing at time t , and vector X k ( t ) represents the information associated with the stock dividend k (that will be paid at time t k ) . For all k = 1 . . .n, Dtk : Rm ++ R+ is a known mapping, and { X k ( t ) ;t 2 0}, X k E Rm the state variable vector of the dividend Dtk, is a linear diffusion defined on a complete filtered probability space (62, F x k ,P ) . We assume that for all i = 1 . . .m, k = 1 . . . n, { x i k ( t ) ;t 2 0) evolves according to the stochastic differential equation
20
Recent Advances in Stochastic Modeling and Data Analysis
where Wik is a standard Wiener process with the instantaneous correlations dWikdWjl = pik,jldt for all i , j = 1 . . . m, k , 1 = 1 . ..n. Read the above such that Xik is the state variable i of the dividend k (a dividend that will be paid at time t k ) . According to It8’s lemma, the stock price must itself follow the It6 process:
where i ,j = 1 . . .m, k , 1 = 1 . ..n, and where a is the expected price appreciabik E d W i k makes the stock price behave stochastically. tion. The term The rest of the study focuses on just this term.
xi,k
3 3.1
Causes of Stochastic Return Volatility Mappings of State Variable
For simplicity the analysis, let us suppose temporarily that there is only one stochastic state variable driving all the dividends, and that the state variable of the dividends { X ( t ) ;t 2 0 } itself follows the geometric Brownian motion:
d x(t)=ex(t)dt
+ O x ( t ) d w ( t ) , x > 0,
(4)
where 0 E R and 0 E R+ are constants. We will relax assumptions later when we will consider multiple state variables and the alternative processes of state variables. Now the asset price process is in the form
dV = adt
V
av + C-vx -dW ax
= adt
+ OEdW,
where a equals the expected price appreciation and E is the elasticity of the asset price to the state variable. If E is stochastic, so is stock return volatility O&, too. Proposition 1. Suppose that dividends’ state variable x follows geometric Brownian motion. If the elasticity of the asset price t o the state variable is a constant, say (, the asset price V = cxc,where c is a positive constant.
Proof. The solution of the differential equation positive constant. 0
%$ = C is V = cxc,where c is a
Remark 1. The above proposition also says that if the asset price is not in the form of V = cxc,then the elasticity of the asset price to the state variable, and hence also stock return volatility, is not constant.
Cause of Stock Return Stochastic Volatility 21 Example 1 . The above means that the assumption of constant return volatility is invalid, if, for example, the asset price V = clxb + C Z , where b, c 1 , c ~are constants. This is the case if the dividend of time t k ,
If x follows (4), then Now
E
is admittedly stochastic:
av x ax v
--
-
Clxbb C1XbfCZ1
Remark 2. Example 1 has an important economic interpretation. Suppose that the price of output good fluctuates according to (4) and that the dividend of time t k is X ( t k ) - a , where a denotes the production costs of the period. Then return volatility equals cr& and is stochastic. Example 2. Another example of stochastic return volatility could be V = f l ( x ) f z ( x ) . . , where f i , i = 1 , 2 , . . . , is the function of x. This is a case if, for example, the dividend of time tk, Dt, = lnX(tk), when E is stochastic even if discount rate p equals zero and number of dividends n equals one:
+
+.
0
We have now considered transformations that result in return volatility to be stochastic. What kind of dividend transformations do produce constant return volatility? The next proposition clarifies this.
Proposition 2. Suppose that dividends ' state variable x follows geometric Brownian motion. If the dividend of time t has a form Dt, = where a and b are constant numbers, then the elasticity of the asset price to the state variable, E , is constant and equals b. Proof. The proposition is a direct implication of Example 1 with a2 = 0 in (5). 0 3.2
Multiple State Variables
We assume that Dtk
=D(Xk(tk)) =Xlk(tk)
+X2k(tk)
f ' . . f X'TLk(tk)?
(6)
k = 1. . .n, and that the state variables of the dividends { X ( t ) ;t 2 0 } follow the geometric Brownian motion with constant drifts and volatilities :
+
dXilc(t) = QikXilc(t)dt aircXik(t)dWik(t),
(7)
22
Recent Advances in Stochastic Modeling and Data Analysis
where x i k > 0 for all i, k . Now stock price as a cumulative present value of expected dividends can be expressed as
where i = 1. . .m, k = 1. . .n, and where p is the stock’s instantaneous discount rate. The stochastic term in (3) takes the form
i = 1. . . m, k = 1 . . . n. How should we interpret this? What can we say now about stock volatility? Proposition 3. Suppose that there are several (more t h a n one) state variables and that (6), (?’),and (8) hold. T h e n the stock return volatility is a constant if and only zf all the state variables are drived by perferctly correlated W i e n e r processes and volatilities of state variables, cr&, are equal t o each other.
Proof. Without loss of generality, suppose that all dividends depend on the same two state variables] X 1 and X 2 . Technically, for all k = 1.. . n D ( x l ( t k ) ~ X Z ( t k= ) )x l ( t k ) + x Z ( t k ) . These variables follows the geometric Brownian motions driven by the Wiener processes W l ( t )and Wz(t), d W I ( t ) d W z ( t ) = p l z d t . Let W’ be an independent Wiener process with respect to Wl and W2. Then we can write d W z ( t ) = p l z d W l ( t ) d m d W * ( t ) . If aik = cr and plz = 1, then we can then write (9) as follows:
+
=VadW(t),
where d W l ( t ) = d W z ( t ) = d W ( t ) . Thus, the stock price evolves with constant volatility. We must still prove that if return volatility is constant] then all the state variables are drived by perferctly correlated Wiener processes and volatilities of state variables, are equal to each other. Let us do this by showing that if processes are not perefectly correlated or if volatilities are not the equal, return volatility is not constant. Again, suppose that all dividends depend on the same two state vasia,hles, XIand X,. The price of the stock is n
V ( t )=
C { X l ( t )exp((o1
-
P)(tk - t ) )+ ~ 2 ( texp((e2 ) - P)(tk - t ) ) > .
Cause of Stock Return Stochastic Volatility 23 We can then write (9) as follows:
where
i = 1 , 2 . Let &t)dl@(t) = +l(t)dWl(t) + + z ( t ) d W 2 ( t ) , when we find that
$(t)= J$l(t>Z
+ + z ( t > Z + 2PlZ+l(t)+Z(t).
(11)
Therefore, stock price diffusion takes the form
+
dV(t)= aV(t) J(t)V(t)dW(t).
(12)
8
Clearly, the stock volatility is admittedly stochastic even if either 01 = 02 or plz = 0. Note that if return volatility depends positively on the correlation PlZ. 0 I have illustrated the stochastic volatility numerically by generating sample paths for state variablesand supposing that equations (7), (10) and (11) determine state variables, stock price, and volatility. The illustration can be found at http://www.tut.fi/"kanniain/ASMDA/illustration.pdf I assumed two state variables, X , ( t ) and X z ( t ) with a constant correlation and with constant volatilities and drifts. Moreover, I assumed that n dividends will be paid at times (years) 1 , 2 , 3 .. . , and I simulated the time interval ( 0 , l ) (the first year). The illustration shows that stock return volatility may vary considerably over time. Moreover, it also examines the effect of the correlation plz on the price paths concluding that if the correlation increases, the stock price becomes more volatile, and the volatility curve moves upward. If the correlation decreases, also stock volatility decreases, and the volatility curve drops. This is in line with our analytical observation of a positive relation between plz and The result is analogous with portfolio diversification: if st.ate variables do not correlate mut'ually, their fluctuations eliminate each other. Moreover, the numerical illustration argues that the greater (less) the correlation, the greater (less) the stock volatility but with less (greater) fluctuation. T h e result is also quite intuitive. As we can see from the above equations, the mutual proportions of the state variable values clearly affect the level of stock volatility. Suppose that dividends depend on two variables, and that the volatility of the first variable, n1, is less than the volatility of the second variable, 02. If the value of the second state variable, Xz, increases in proportion to the value of the second state variable, X I ,
4.
24
Recent Advances in Stochastic Modeling and Data Analysis
the more volatile state variable takes room, and stock volatility increases. Similarly, if X I increases in proportion to X z , stock volatility decreases. Obviously also the greater (less) the correlation between state variables, the less (greater) their mutual proportion changes over time. Therefore, if state variables evolve with different volatilities and a low, or even negative mutual correlation, stock volatility may fluctuate substantially. We could interpret this result economically in that if a business depends on homogenous (heterogeneous) factors (in the sense of statistical dependency), its volatility is high (low) and does not (does) vary much. I also illustrated the effect of discount rate to volatility and its fluctuation. Here the effect of the discount rate on stock volatility can be either positive or negative. In addition, the discount rate has a great effect on how stock volatility varies over time.
3.3
Alternative Characterizations of State Variables Processes
Finally, we consider alternative processes of the state variables. Suppose that a i k ( t , x i k ) = q i k ( x i , - x&(t)) and b i k ( t , X z k ) = U i k , in which case the state variable { X i k ( t ) ;t 2 0}, z = 1 . . . m, k = 1 . ..n evolves according to the stochastic differential dxik(t)
= qik
( X i k - x i k ( t ) ) dt
+uikdWik(t),
where V i k , X i k , and f f i k are constant numbers for all z, k . This is the so-called mean-reverting process. Now, because for some 7 > t (see, for example, [Dixit and Pindyck, 1994, p. 741)
+ ( X i k ( t ) - X i k ) eXP ( - q i k ( T - t ) ) the stock price with mapping Dt, = X I k ( t k ) f X 2 k ( t k ) + ' . + X m k ( t k ) V ( t )= { x i k + ( x i k ( t ) - X i k ) exp (-'%k(tk t ) ) }exp ( - P ( t k t)). Et
[ x i k ( T ) ] = XEk
1
'
x
-
-
i,k
Stock return volatility is now unquestionably stochastic since the stochastic term in (3) takes the form
Note that stock volatility would be stochastic even if f f i k = cr and W i k ( t ) = W ( t )for all i = 1 . ..m, k = 1 . .. n, t > 0, in which case volatility would be equal to
Remark 3. The reason here is that the variance rate does not grow with x. Therefore, if dividends are driven by such a process, stock volatility remains unquestionably stochastic with linear mappings of state variables.
Cause of Stock Return Stochastic Volatility 25
4
Conclusions
The starting point of this study was that dividends are driven by state variables and that investors monitor the state variables and continuously revise their beliefs regarding to stock price. The paper studied how the mappings of state variables are related stochastic return volatility. Moreover, also multiple state variables were explored. The main result is that the stock return volatility is admittedly stochastic if future dividends are affected by more than one stochastic state variable. Thus, the paper affirms the invalidity of the geometric Brownian motion as models of stock price. We observed that the correlation between state variables has an effect on volatility dynamics according to the greater (less) the correlation, the greater (less) the stock volatility, with less (more) fluctuation over time. In addition, we found that the discount rate affects volatility and its fluctuation positively or negatively.
References [Black, 19761F. Black. Studies of stock price volatility changes. Proceedings of the 1976 meetings of the American Statistical Association, Business and Economic Statistics Section, pages 177-181, 1976. [Christie, 1982lA. A Christie. The stochastic behavior of common stock variances: Value, leverage and interest rate effects. Journal of Financial Economics, 10:407-432, 1982. [Dixit and Pindyck, 1994lA. K. Dixit and R. S. Pindyck. Investment Under Uncertainty. Princeton, New Jersey, Princeton University Press, 1994. [Hamilton and Lin, 19961J . D. Hamilton and G. Lin. Stock market volatility and the business cycle. Journal of Applied Econometrics, 11:573-593, 1996. [Mascaro and Meltzer, 1983lA. Mascaro and A. H. Meltzer. Long- and short-term interest rates in a risky world. Journal of Monetary Economics, 12:485-518, 1983. [Merton, 19741R. C. Merton. On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance, 29:44%470, 1974. [Nielsen, 19991L. T . Nielsen. Pricing and Hedging of Derivative Securities. Oxford University Press, 1999. [Officer, 1973lR. R. Officer. The variability of the market factor of the new york stock exchange. Journal of Business, 46:434-453, 1973. [Schwert, 19891G. W. Schwert. Why does stock market volatility change over time? Journal of Finance, 44:1115-1153, 1989. [Schwert, 19901G. W. Schwert. Stock market volatility. Financial Analyst Journal, 46:23-34, 1990.
On a class of backward stochastic differential equations and applications to the stochastic resonance Romeo Negrea' Department of Mathematics Politehnica University of Timisoara P-ta Victoriei 2, 300006 Timisoara ROMANIA (e-mail: negreabath. uvt .ro)
Abstract. In this paper, we intend to prove a result on the solvabillity of a class of backward stochastic differential equations by the McShane type. We will considered some more general conditions by the coefficient functions and prove a result on the existence and the pathwise uniqueness by the Nagumo type, extended the Athanassov results ([Athanassov, 19901) for the ordinary differential equations. In final, we study the control of some electronic circuits in the presence of the stochastic resonance. Keywords: backward stochastic differential equations, adapted solution, pathwise uniqueness, belated integrals, stochastic resonance, signal-to-noise ratio.
1
Introduction
Noise in dynamical system is usually considered a nuisance. However, in certain nonlinear systems, including electronic circuits and biological sensory systems, the presence of noise can enhance the detection of weak signals. The phenomenon is termed stochastic resonance and is of great interest for electronic instrumentation. The essential ingredient for the stochastic resonance is a nonlinear dynamical system, which typically has a period signal and noise at the input and output that is a function of the input as well as the internal dynamics of the system. The nonlinear component of the dynamical system is sometimes provided by a threshold which must be crossed for the output to be changed or detected. A nonlinear system is essential for stochastic resonance to exists, since in a system that is well characterized by linear response theory, the signal-to-noise ratio at the output must be proportional to the signal-to-noise ratio at the input. Engineers have normally sought to minimize the effect of noise in electronic circuits and communication systems. Today, however, it is acknowledged that noise or random motion is beneficial in breaking up the quantization pattern in a video signal, in the dithering of analog to digital converters, in the area of Brownian ratchets, etc. 26
O n a Class of BSDE 64 Applications to the Stochastic Resonance
27
In the field of control, we usually regard y(.) as an adapted control and
x(.) as the state of the system. We are allowed to choose an adapted control y(.) which drives the state x(.) of the system to the given target X at time t = 1. This is so-called reachability problem. So in fact we are looking for a pair of stochastic processes { x ( t ) y(t), , 0 2 t 5 1) with values in R x R which is Ft-adapted and satisfies the above equation. Such pair is called an adapted solution of the equation and was introduced in [Pardoux and Peng, 19901. In the electronic circuits with stochastic resonance we have some nonlinear system, therefore, must to be considered more general coefficient functions for a backward stochastic differential equations - non-linear and maybe non-lipschitz. On the other hand, in practice, the additional input noise (which is responsible for the resonance) and the external perturbations belong to a large class of noises, then the stochastic calculus by the McShane type (introduced in [McShane, 19741) can be a better approach for the mathematical models of some phenomena. In this paper, we intend to prove a result on the solvabillity of a class of backward stochastic differential equations by the following type:
where { z j ( t ) , 0 5 t 5 l}, j = 1 , 2 , . . . , T is a stochastic process defined on the probability space (R, .F, P ) with the natural filtration {Ft,0 5 t 5 l} and X is a given F1-measurable random variable such that EIX12 < 00. Moreover, f is a mapping from R x [0,1] x R x R to R which is assumed to be P @I B @I B \ B-measurable, where P is the 0-algebra of Ft-progressively measurable subsets of R x [0,1]. Also g is a mapping from 0 x [0,1] x R to R which is assumed to be P x B \ B-measurable. All stochastic integrals are belated or McShane integrals. A version of the equation (1)which has the most important mathematical properties is its canonical form:
2
Preliminary results
In 1974, E.J. McShane, introduced so called belated integrals and stochastic differentials and differential systems which enjoying the following three
28
Recent Advances in Stochastic Modeling and Data Analysis
properties: inclusiveness, consistency and stability. McShane's calculus had proved to very valuable in modeling and is finding applications in physics, engineering and economics ([McShane, 19741, [McShane, 19751). Therefore, in this section we recall some specifically results of the McShane stochastic calculus. Let (Q,.F,P) be a complete probability space and let { F t , 0 5 t 5 u } be a family of complete a-subalgebras of .F such that 0 5 s 5 t 5 a then
Fs c 3 t .
Every process denoted by z with differents affixes will be a real valued second order stochastic process adapted to {Ft,0 5 t 5 u } (i.e. z ( t ) is .?-measurable for every t E [0, u]) and
IE[(z(t)- z(s))"/Fs]l i K ( t - S )
5 a, m = 1,2,4, for a positive constant K having as. continuous sample functions (and we say that the process satisfies a a.s., whenever 0 5 s 5 t
K-condition). It is known (see [McShane, 19741) that if f : [O,u] + L2 is a measurable process adapted to the Ft and if t ---t Ilf(t)lI is Lebesgue integrable on [O, a], then if z1 and z2 satisfy a K-condition, the McShane integrals
exist and the following estimates are true ra
where C = 2Ku4
3
+Ki.
Theoretical results
Some results in weaker conditions than the Lipschitz conditions for the coefficient functions of a backward stochastic differential equation are given in [Mao, 19951, [Lepeltier and San Martin, 19981, [Lepeltier and San Martin, 20021, [Kobylanski, 20001 or [Constantin, 20041, [Negrea, 20061 and many others. A very interesting applications was given, for example, in [Constantin, 20051, [Ladde and Sambandham, 20031, [Srinivasan and Udayabhaskaran, 19821, In [Athanassov, 19901 is proved a uniqueness theorem of Nagumo type for the Cauchy problem generalizing several known uniqueness theorems and sufficient conditions to guarantee the convergence of the Picard successive
O n a Class of BSDE &' Applications to the Stochastic Resonance
29
approximations for the ordinary differential equations. Stochastic generalizations of the results of Athanassov for (forward) stochastic differential equations are given in[Constantin, 1995][Constantin, 19961 [Constantin, 19981, [Constantin and Negrea, 20041 or [Negrea, 20031 and others. But it is known that the uniqueness of the solution of an initial value problem of Cauchy and the convergence of successive approximations are logically independent, i.e. the uniqueness of the solution does not ensure the convergence of successive approximations nor is the converse true. For the equation (4) we consider the following hypotheses: i) gj and its partial derivations are P 8 B 8 B measurable functions; ii) gj(.,O,O) E M 2 ( ( 0 , 1 ) , R ) ,g ( . , O , O ) E M2((0,1),R)and %(.,O,O) E
M 2 ( ( 0 ,I ) , R); iii) there exists u(t) a continuous, positive and derivable function on 0 < t 5 1 with u(0)= 0, having nonnegative derivative u'(t) E L([O,l]),with u'(t) + 00, t + Of such that if p is any one of the functions g j , relation (2)) we have
hjk
(as in
for al121,~2,yl,y2E R, 0 I t 5 1, a n d A = (r+1)C2+(r2+1)C2+(2r2)C2.
iv) with the same functions u ( t )as above,
for all z , y l , y 2 E R,0
< t 5 1.
In [Negrea and Caruntu, 20071 is given the result on the existence and uniqueness of the solution of the equation (4). More exactly, we have the following theorem:
Theorem 3.1. Let be gj and h j k satisfying the above hypotheses for any j , k = 1,.. . , T and X E L2(f2,F1,P, R),then there exists a unique pair (z,y) E M 2 ( ( 0 , 1 ) , R )which satisfy the equation (4) in the canonical form for [S, 11, for any positive constant 6. Now, we will present some results on the stability properties of the solution of the equation (1). We consider families of backward stochastic integral equations
30
Recent Advances in Stochastic Modeling and Data Analysis
whit X E A- a open and bounded set
c R".
In similar way as in [Negrea, 20061 and using the properties (3), (4), (5) and (6) it easy to prove the following two results:
Theorem 3.2.If, f o r any X E A, the coeficient functions fx a n d g x satisfy the hypothesis (i)-(iv), then the family (4) has a unique solution (xx,y x ) E W(0,11). Moreover, if IXx,, - Xxl -+ 0, m -+ 00, then, we have that
Theorem 3.3.h the hypothesis (i)-(iv), XxoI = 0 , then lim Izx
X-Xo
-
we have that if limx+xo IXx
xxo12 = 0 , and lim Iyx X-Xo
-
-
yxOl2= 0 ,
on any compact subset of the domain (0,1], where cp is any functions f or g.
4
Applications to the Stochastic Resonance
In this section we study some applications of a general class of McShane backward stochastic differential equations described above. For a better understanding of the notion of stochastic resonance, we consider the following mechanical experiment - a ball is pushed with a force F on a non-linear path, and let to consider the following three cases: a) Non-perturbed case - if the force F is strong enough and there are not external perturbations, then the ball arrives at the final point. b) Perturbed case - if the there are some external perturbations, the ball do not rise "'the first pick", therefore is increase the force F and the ball pass to the first hill, but if there are more external perturbations the ball can not pass to the second ball and it not return to the initial point for a new increasing of the force F .
O n a Class of BSDE 8 Applications to the Stochastic Resonance
31
c) Perturbed case with a "jumper ball" - the force F pushes not a jumper ball, a ball with a "noise". If there is a "good noise", the ball pass to the all external perturbations and arrive to the final point. A model of one-dimensional nonlinear system that exhibits stochastic resonance is the damped harmonic oscillators with the Langevin equation of motion: rnji.(t)
+ yk(t) =
dU(x) dx
--
+ fi[(t)rnji.(t) + yk(t) = --dU(x) + dE[(t) dx
This equation describes the motion of a particle of mass m moving in the presence of friction . The restoring force is expressed as the gradient of some bistable or multi-stable potential function U(x). In addition, there is an additive stochastic force [(t) with intensity D , and, in generally, it supposed as been a white Gaussian noise. U ( x ) = - a$+b$ In the case of bistable system, in [Gammaitoni et al., 19981, [Harmer et al., 20021 the potential function is a simple symmetric function and, adding a period signal and considering case of time dependent system U(x, t) = U ( z )- Axsin(w,t) = -a-
22
2
+ b-x44
-
Axsin(w,t)
where A and are the amplitude and the frequency of the periodic signal, respectively. More models for the electronic circuits with a stochastic resonance are discrete. In general, they used a state space time series models given by the bellow expressions z n = f (xn-1) un Yn = h(xn) wn
{
+
+
where f and h are non-linear functions and un (input noise) and wn (external noise) are respectively Gaussian or non-Gaussian noises. About these models there is a simple observation: we can not say exactly if the external perturbation is present just at the discrete moments. A continuous model appears as more adequate. This approach is possible just using the theory of stochastic differential equations. Because, the control of the input noise is the key of the benefit of the stochastic resonance, we use the backward stochastic differential equations. The problem, which frequency appear in practice, is the value of initial state 20. This value is "proposed" but in some non-standard external conditions, this make an discontinuity of the simple path of the process {z(t)} After this time (t=O), the adapted control {y(t)} works and we have a continuity of {x(t)} . Therefore, is necessary to consider some general coefficient functions and good stability properties of the solutions. On the other hand, the stochastic resonance make possible a control of the electronic circuits in some external stochastic perturbations by controlling the adapted process {y(t)} (see [Gammaitoni et al., 19981, [Harmer et al., 20021, [Calvo and Chialvo, 2006]), but the control appears as the result of filtering.
32
Recent Advances in Stochastic Modeling and Data Analysis
About above models we make the following remarks: a) in choosing the type of stochastic processes that we shall use a models of the noises we meet a dilemma. On the one hand, there is no physical basis for considering function with properties such as non-differentiability and on the other hand, the involved process must be of a kind that we can manage mathematically. b) in almost approaches, the disturbance is supposed to be Gaussian white noise, but, is known that a electrical system cannot support more than some limited current or voltage difference without destruction. Therefore, the Gaussian white noise which is a special derivate of Brownian motion (in the sense of distributions) must replace or combine with some other noises, more smoothed. For some numerical results, we consider a Chua’s type electronic circuit (see [Harmer et al., 20021, [Beglund and Gentz, 20021)with a discontinuity at time t = 0. More precisely we consider the coefficient functions f ( t , z, y) =
--t$z3+tkz+sin(27rt/T),
g(t,z,y) = y
4-t:, ~ ( 0 )
= 0,
T =X
= 1, z1 =
W.
For the sequences of successive approximations of { ~ ( t and ) } {y(t)} we have the following simple path XI
Y
Fig. 1. Simple path of the signal and the noise. In conclusions we can see the effect of the adapted process in the bistable regime of the state process { ~ ( t ) } .
References [Athanassov, 19901Z. S. Athanassov. Uniqueness and convergence of successive approximations for ordinary differential equations. Math. Japonica, 53:351-467, 1990. [Beglund and Gentz, 2002lN. Beglund and B. Gentz. A sample path approach to noise-induced synchronization: Stochastic resonance in double well potential. Ann. Appl. Probab., 12:1419-1470, 2002. [Calvo and Chialvo, 200610. Calvo and D.R. Chialvo. Ghost stochastic resonance in an electronic circuit. Inter. Journ. of Bifurcation and Chaos, 16:731-735, 2006. [Constantin and Negrea, 20041Gh. Constantin and R. Negrea. An application of schauder’s fixed point theorem in stochastic mcshane modeling. J. Fixed Point Theory, 5:37-52, 2004.
O n a Class of BSDE
tY Applications to the Stochastic Resonance
33
[Constantin, 1995lA. Constantin. Global existence of solutions for perturbed differential equations. Annuli d i Mat. Pura ed Appl., IV:237-299, 1995. [Constantin, 1996lA. Constantin. On the existence and pathwise uniqueness of solutions of stochastic differential equations. Stochastic and Stochastic Reports, 56~227-239, 1996. [Constantin, 1998lA. Constantin. On the existence and uniqueness of solutions of mcshane type stochastic differential equations. Stoch.Ana1. Appl., 16:217-229, 1998. [Constantin, 20041Gh. Constantin. The uniqueness of solutions of perturbed backward stochastic differential equations. J. Math. Anal. Appl., 300:12-16, 2004. [Constantin, 20051Gh. Constantin. An application of the stochastic mcshane’s equations in financial modeling. In J. Janssen and P. Lenca, editors, Proceedings of Applied Stochastic Models and Data Analysis, pages 917-925, 2005. [Gammaitoni et al., 1998lL. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni. Stochastic resonance. Rev. of Modern Physics, 70:223-287, 1998. [Harmer et al., 2002lG.P. Harmer, B.R. Davis, and D. Abbot. A review of stochastic resonance: Circuits and measurement. IEEE Trans. on Instr. Measur., 51:299309, 2002. [Kobylanski, 20001M. Kobylanski. Backward stochastic differential equations and partial differential equations with quadratic growth. The Annals of Probability, 28:558-602, 2000. [Ladde and Sambandham, 20031 G.S. Ladde and M. Sambandham. Stochastic versus Deterministic Systems of Differential Equations. Marcel Dekker, Inc., New York, 2003. [Lepeltier and San Martin, 19981J.-P. Lepeltier and J. San Martin. Existence for bsde with sperlinear-quadratic coefficient. Stoch. Stoch. Reports, 63:227-240, 1998. [Lepeltier and San Martin, 2002]J.-P. Lepeltier and J. San Martin. On the existence or non-existence of solutions for certain backward stochastic differential equations. Bernoulli, 8:123-137, 2002. [Mao, 1995lX. Mao. Adapted solutions of backward stochastic differential equations with non lipschitz coefficients. Stoch. Proc. and Their Appl., 58:281-292, 1995. [McShane, 1974lE.J. McShane. Stochastic Calculus and Stochastic Models. Academic Press, New York, 1974. [McShane, 1975lE.J. McShane. Stochastic differential equations. J.Multivariate Analysis, 5:121-177, 1975. [Negrea and Caruntu, 2007lR. Negrea and B. Caruntu. On certain class of backward stochastic differential equations by mcshane type. Analele Univ. Timisoara, Ser. Mat.-Inf, 45:225-234, 2007. [Negrea, 20031R. Negrea. On the uniqueness and convergence of successive approximations for a class of stochastic differential equations. Analele Univ. Bucuresti, Ser. Mat., 52~225-234, 2003. [Negrea, 2006lR. Negrea. On the existence and uniqueness of solutions for certain backward stochastic differential equations. to appear, 2006. [Pardoux and Peng, 19901E. Pardoux and S.G. Peng. Adapted solution of a backward stochastic differential equation. Systems & Control Letters, 14:55-61, 1990. [Srinivasan and Udayabhaskaran, 1982lS.K. Srinivasan and S. Udayabhaskaran. Modeling and analysis of dynamical systems subject to discontinuous noise processes. Journ. Math.Phys.Sci., 16:415-430, 1982.
CHAPTER 2
Distributions
An application of the Extended Waring distribution to model count data variables Jose Rodriguez Avi' , Antonio Conde S h c h e z l , Antonio Jose Shz-Castillo', and Ma Jose Olmo Jinienez' Department of Statistics and Operations Research University of Ja6n 23071 Jakn, Spain (email: jravioujaen.es) Abstract. An extension of the univariate Waring distribution is proposed as an alternative to the U G W D for overdispersed count data in those cases in which the
parameter estimates do not allow the properties of the U G W D to be used, such as the partition of the variance. Specifically, this model is applied to study the set of variables number of hotels per municipality in the Autonomous Region o n Andalusia (Spain) f r o m 1990 t o 2003. Keywords: Waring distribution, Number of hotels, Overdispersed count data.
1
Introduction
To study the impact of tourism in the economy, an interesting variable can be the number of hotels per municipality within a particular geographical framework. Specifically, the aim of this work is to find a probabilistic model to describe, in an appropriate way, the variable X, number of hotels per municipality in the Autonomous Region of Andalusia (Spain) from 1990 to 2003. These data have been compiled from the data bank of the System of Multi territorial Information of Andalusia (SIMA, by its Spanish acronym, [IEA, 20061). The observed frequencies for the 14 variables appear in Table 1.
For all years we obtain discrete count variables that have a minimum in 0 which is also a modal value, whereas the maximum is a very high value
that increases from one year t o another. A summary of the main descriptive characteristics of the 14 variables is shown in Table 2. The aggregation index shows that the 14 variables have strong overdispersion, so the variability is not only due t o randomness. Firstly, an important factor that influences the observed variability is the size of the municipality. However, this is not the only deciding factor. Thus, cities with great tourist importance, such as Mdaga or Granada, have greater number of hotels than Sevilla, the capital of Andalusia and the most populated. The same happens with coastal towns compared with inland towns, etc. Other factors to 35
36
Recent Advances in Stochastic Modeling and Data Analysis - - 91 92 13t i2: i0: 62 62 71 18 24 27 10 12 16 13 13 12 6 8 9 4 7 9 4 2 3 0 1 3 2 2 0 0 3 2 1 0 2 3 1 2
qEi - 90 0 1 2 3 4
5 6 7 8 9 10 11 12 13 14 f15
1 0
7
-
1 1 1 0 9 8 -
-
-
)3 99 14 !7 15 14 7 8 4 4 0 2 2 1 1 1 9 -
9 4 88 34 17 14 14 11 6 2 5 3 1 3 1 1 0 9 -
3 5 81 36 19 14 15 9 8 3 5 3
36 87: 39
- - 37 - 98 - 99 16i 15; l4( 3 1 O i .0; 3 1 32 29 35 14 16 20 18 14 14 10 13 9 9 4 5 8 8 11 9 3 4 5 6 4 4 5 5 5 2 1 1 1 3 4 3 3 1 2 3 1 0 3 2 3 1 2 1 2 1 3 3
- )1 12 - 13 1 2 01 8E 24 24 38 i2 17 5 1 19 13 14 9 11 16 9 4 5 7 LO 6 1 5 6 8 7 6 4 4 4 3 4 4 3 1 1 2 2 4 3 1 1 0 2 3 0 0 1 3 0 0 1 1 0 4 3 2 2 11 13 14 16 LO 18 - LO - 10 - - - 17 - -
-
00 i2: -2: 41 16 13 6 9
Table 1. Observed frequencies for the hotel data from 1990 to 2003
be taken into account are the financial development of the municipality, the business initiative, etc.
Tf=EF
Range Aggregation in( 1991 767 0.8292 13.6227 1992 768 0.9440 16.1492 1993 768 0.9544 16.3899 1994 769 1.0117 19.5980 1995 769 1.0351 16.8479 1996 769 1.1235 19.5309 1997 769 1.1599 20.0537 1998 769 1.1274 18.6001 1999 769 1.1938 20.1484 2000 769 1.3199 22.8808 2001 770 1.3844 23.8886 2002 770 1.4909 27.0162 2003[77011.61561 31.0912
48 54 57 67 56 59 60 59 60 67 68 71 78
16.4287 17.1072 17.1730 19.3714 16.2766 17.3840 17.2892 16.4982 16.8775 17.3353 17.2556 18.1207 19.2444
Table 2. Descriptive summary
An Application of the Extended Wuring Distribution 37
2
Negative binomial model
Since data have overdispersion, a model that could be proposed to describe them is the negative binomial model, N B , that can be seen as the mixture
= Poisson ( A ) AA G a m m a ( a ,k ) This means that the average number of events is not constant but follows a gamma distribution. In this case, the total variance can be split into two components:
+
V ~( X T ) = E A ( V ~( T X l n ) ) V W A( E ( X ~ A .) )
(2)
The second term of (2) represents the variability due to sources of variation that cause overdispersion, whereas the first term reflects the variation due to randomness. The N B ( a , p ) distribution may also be seen as belonging to the Gaussian hypergeometric distribution family, G H D , ([Johnson et al., 20051) because its probability generating function (p.g.f.) is given by:
Table 3 includes the results obtained using the N B model for the hotel data. The estimates have been obtained by the maximum likelihood method.
NB(a,p) x' p-value log-likelihood 1990 (0.0799, 0.0996) 13.1479 0.0686 -626.2 -686.0 1991 (0.0881, 0.0961) 10.2694 0.2466 -760.1 1992 (0.1032, 0.0986) 10.1610 0.3366 -768.3 1993 (0.1057, 0.0997) 10.1809 0.3360 -803.6 1994 (0.1144, 0.1016) 16.7173 0.0533 -827.3 1995 (0.1214, 0.1050) 15.2138 0.0827 -860.0 1996 (0.1242, 0.0995) 15.0755 0.1293 -880.2 1997 (0.1287, 0.0999) 15.7031 0.1085 1998 (0.1418, 0.1117) 33.0476 0.0000 -890.1 1999 (0.1466, 0.1094) 28.6091 0.0014 -919.4 2000 (0.1632, 0.1100) 46.2239 0.0000 -928.6 2001 (0.1786, 0.1143) 46.7014 0.0000 -1026.0 2002 (0.1847, 0.1102) 49.3454 0.0000 -1065.7 2003 (0.1945, 0.1074) 56.3517 0.0000 -1112.1
Year
Table 3. MLE, X2-goodness of fit test and log-likelihood for the hotel data using
the N B model
It should be emphasized that only in the first years - until 1997 - the fits are acceptable whereas in the last years the fits are very bad.
38
Recent Advances in Stochastic Modeling and Data Analysis
3
Fit by a U G W D
The univariate generalized Waring distribution, U G W D , with parameters ( a , k and p ) has also been proposed as a valid model in the case of overdispersion ([Johnson e t al., 20051). It is also a distribution that belongs to the G H D family since its p.g.f. can be expressed as:
This distribution can be obtained from a N B ( a , p ) when the parameter p is not constant but follows a BetaI(k, p) distribution, that is
( p')
P o i s s o n ( A ) A Gamma a , -
Beta1 ( p ,k ) .
(5)
The results obtained fitting the U G W D model and applying the maximum likelihood method ([Rodriguez Avi et al., 20031) are shown in Table 4. It can be seen that the values of the X2-statistic are lower than those provided by the N B fit for all years, and values for the log-likelihood are greater. Year
-
UGWD(a,k,p)
XL 1990 (0.8137, 0.2707, 1.1087) 7.0334 1991 (5.8800, 0.1296, 1.8618) 3.5696 1992 (5.5370, 0.1562, 1.8669) 4.2428 1993 (4.9568, 0.1651, 1.8020) 3.8855 1994 (0.8623, 0.4219, 1.1977) 7.8420 1995 (1.3074, 0.3299, 1.2557) 6.2382 1996 (3.7016, 0.2151, 1.6272) 5.7105 1997 (0.5972, 0.6822, 1.1723) 5.8098 1998 (0.4958, 1.0040, 1.3141) 13.677( 1999 (0.6326, 0.7998, 1.2907) 7.9839 2000 (0.7727, 0.7727, 1.3377) 8.2827 2001 (0.8332, 0.8331, 1.4081) 7.3912 2002 (0.8636, 0.8581, 1.4044) 10.191: 2003 (0.8957. 0.9029. 1.4158) 12.873(
p-value log-likelihood AIC 0.2182 0.7347 0.7515 0.7929 0.2449 0.5122 0.6796 0,6685 0.0572 0.3340 0.4064 0.4951 0.3352 0.1684
-621.71 -681.18 -753.75 -761.52 -794.30 -819.98 -851.34 -872.30 -875.96 -904.80 -963.00 -1003.7 -1042.0 -1084.1
76.66 83.95 84.12 83.81 90.12 91.52 91.52 93.46 96.35 96.35 112.04 106.40 116.67 119.10 ~
Table 4. MLE, X2-goodness of fit test, log-likelihood and AIC for the hotel data using the U G W D model
Comparing Table 3 and Table 4, it is clear that the U G W D model improves the fits that provides the N B model in all cases. Since the U G W D and the NB are not nested models the Vuong test ([Winkelmann, 20031, pg 109) has been carried out as an extension of the likelihood ratio test. The null hypothesis indicates that the two models are
A n Application of the Extended Waring Distribution
39
equivalent versus the alternative hypothesis that the U G W D model is preferable t o the N B one: ffo : E (LUGWD(%GWD) - L B N ( @ B N=) )0
H i :E(LUGWD(~GGW -D L B) N ( @ B N > ))0
(6) (7)
The test statistic is given by
where
is the variance of the individual differences in log-likelihood divided by n. So, under the null hypothesis, the L R statistic converges in distribution to a standard normal distribution. The results of the test for each year appear in Table 5. The p-value is less than 0.1 in all years, except the year 1990, and decreases, specially, from 1998. Therefore, the U G W D model may be considered, in general, to be better than the N B model. Year Statistic 1990 1.0206 1991 1.6696 1992 1.7954 1993 1.7986 1994 1.4743 1995 1.3395 1996 1.8765 1997 1.2916 1998 2.2484 1999 2.2799 2000 2.8479 2001 3.1587 2002 3.2674 2003 3.6413
7-valuc 0.1537 0.0475 0.0363 0.0360 0.0702 0.0902 0.0303 0.0982 0.0123 0.0113 0.0022 0.0007 0.0005 0.0001
Table 5. Vuong test between the U G W D and the N B
Many studies have demonstrated the main properties of the U G W D ([Irwin, 19681, [Irwin, 19751, [Xekalaki, 1983133, [Xekalaki, 1983a], among others). Particularly, in order to interpret the data variability, the decomposition
40
Recent Advances in Stochastic Modeling and Data Analysis
a s a mixture of three distributions provides a partition of the variance into three components ([Irwin, 19683):
when p > 2. The first of these is related to random factors, the second to the variability due to external factors that affect the population (liability), and the third to the differences in the internal conditions of the individuals (proneness) ([Irwin, 19681). However, p^ < 2 in the 14 years studied, so the fitted distributions have infinite variance.
4
Fit by an Extended Waring distribution
Trying to solve the aforementioned problem, the hotel data will be modelled by a tetraparametric G H D of type I ([Rodriguez Avi et al., 20031) whose p.g.f. has the expression
with a , P , y > 0, 0 < A 5 1 and 0 < y. This distribution is called Extended Waring distribution, E W D , ([Rodriguez Avi et al., 20061) and includes, as limit cases, the U G W D ( u ,k , p ) when a = a , P = k , y = a k p and X = 1 and the N B when y is equal to a or /3 and X = 1 - p . Moreover, like the U G W D , the E W D can be expressed as a mixture. Specifically:
+ +
(
Poisson ( A ) A G a m m a a , A
- )' 1-X(1-P)
)
A P
G B e t a (y - a - PIP,a , A ) ,
(12) where G B e t a (y - a - p, p, a , A) denotes a generalization of the Beta distribution whose density function is
with y > /3 > 0, a > 0 and 0 < X 5 1 ([Rodriguez Avi et al., 2007al). A similar continuous distribution appears when [Rodriguez Avi et al., 2007bl develop a generalized beta binomial distribution, GBB. Once again the MLE are computed. A summary of all the fits is shown in Table 6 where it can be seen that the results are very similar to those obtained by the U G W D . Taking (12) into account, the variance can also be split into three compcnents: randomness, liability and proneness. X < 1 in all cases, so the fitted model has finite variance and the three components can be computed as it is shown in Table 7. h
A n Application of the Extended Waring Distribution 41 AIC Year EWD(%P;7;4 xs p - d u e log-likelihood -620.22 77.46 1990 (0.3682,0.3672,1.4041,0.9736) 8.0857 0.1516 84.41 -680.94 1991 (2.6136,0.1464,3.8107,0.9821) 3.8337 0.5726 -753.58 86.50 1992 (3.1237,0.1707,4.5378,0.9870) 4.6440 0.5902 -761.28 85.16 1993 (2.2502,0.1919,3.5235,0.9840) 4.0410 0.6711 -793.15 89.73 1994 (1.1743,0.2906,2.4461,0.9852) 5.3354 0.5016 -818.62 91.55 1995 (0.5193,0.4994,1.7711,0.9742) 4.1903 0.5554 91.07 1996 (0.9950,0.3260,2.1547,0.9791) 4.9794 0.6625 -850.67 -870.40 92.96 1997 (0.9950,0.3415,2.1749,0.9792) 3.2506 0.8609 96.30 -875.10 1998 (0.6287,0.6284,2.2798,0.9859) 12.5357 0.0510 -903.93 96.30 1999 (0.6481,0.6469,2.3196,0.9863) 6.7368 0.4568 112.95 -962.42 2000 (0.7239,0.7239,2.5868,0.9905)12.7388 0.0787 -1003.1 107.82 2001 (0.7964,0.7955,2.8477,0.9930) 7.7258 0.3574 118.41 -1041.7 2002 (0.8238,0.8235,2.9047,0.9936)12.3208 0.1375 120.73 -1083.9 2003 (0.8692,0.8685,3.0368,0.9952) 12.8017 0.1189
Table 6. MLE, X2-goodnessof fit test, log-likelihood and AIC for the hotel data using the E W D model
G
Year Total variance b Randomnes 3.12% 3.69% 2002 40.4804 3.94% 2001 35.2344 2000 31.4221 4.19% 4.76% 1999 25.0598 4.88% 1998 23.1130 22.1181 5.25% 1997 5.25% 1996 21.3852 5.87% 1995 17.5759 1994 21.4973 4.70% 1993 18.4643 5.17% 1992 5.02% 18.7985 1991 5.30% 15.6626 1990 5.55% 12.9923
6 Liabilitj '0 Pronenesz 50.96% 55.83% 56.55% 58.80% 61.34% 61.81% 50.55% 50.45% 65.94% 49.28% 64.14% 70.71% 67.28% 71.96%
45.91% 40.48% 39.31% 37.01% 33.90% 33.31% 44.20% 44.30% 28.19% 46.02% 30.69% 24.28% 27.42% 22.49%
Table 7. Partition of the variance for the hotel data
5
Conclusions
The E W D distribution is more flexible than the U G W D distribution to model overdispersed count data sets where the excess dispersion is assumed to be due to external factors (modelled by the gamma ditribution) and internal factors (modelled by the beta distribution). So, there are data sets that are adequately modelled by the E W D but not by the U G W D ([Rodriguez Avi et al., 2007al). When the fits provided by both models are acceptable and similar, the best model is the U G W D since it has less number
42 Recent Advances in Stochastic Modeling and Data Analysis
of parameters, according to t h e principle of parsimony. However, t h e estimation results sometimes do not allow t h e inherent properties of t h e U G W D to b e exploited, in particular, t h e partition of t h e variance. I n these cases, t h e E W D model is proposed as an excellent alternative to t h e U G W D model. This is the example included in this work.
References [IEA, ZOO6]Institute of Statistics of Andalusia IEA. System of Multi territorial Information of Andalusia (SIMA). Web access: http://www.juntadeandalucia.es/institutodeestadistica/sima/indexZ.htm, 2006. [Irwin, 196815. 0. Irwin. The generalized waring distribution applied to accident theory. Journal of the Royal Statistical Society A , 131:205-207, 1968. [Irwin, 197515. 0. Irwin. The Generalized Waring Distribution. Part I. Journal of the Royal Statistical Society A , 138:18-31, 1975. [Johnson et al., 2005lN. L. Johnson, A.W. Kemp, and S. Kotz. Univariate Discrete Distributions. Wiley, New Jersey, 2005. [Rodriguez Avi et al., 2003]J. Rodriguez Avi, A. Conde Shchez, A.J. S&ez Castillo, and M.J. Olmo Jimknez. Estimation of parameters in gaussian hypergeometric distributions. Communication in Statistics, Theory and Methods, 32:11011118, 2003. [Rodriguez Avi et al., 2007alJ. Rodriguez Avi, A. Conde Sanchez, A.J. S&ez Castillo, and M.J. Olmo Jimknez. A new extension of the waring distribution. Computational Statistics and Data Analysis, in press, 2007. [Rodriguez Avi et al., 2007blJ. Rodriguez Avi, A. Conde Sbchez, A.J. S&ez Castillo, and M.J. Olmo Jimknez. A generalization of the betabinomial distribution. Journal of the Royal Statistical Society, Series C: Applied Statistics, 56:in press, 2007. [Rodriguez Avi et al., 2006lJ Rodriguez Avi, A. Conde Shchez, A.J. S&ez Castillo, and M.J. Olmo Jimhez. Extended waring bivariate distribution. In Callej6n J . Herrerias J. Herrerias, R., editor, Distribution Models Theory, pages 221-232, 2006. Econometric Analysis of Count Data. [Winkelmann, 2003lR. Winkelmann. Springer, Heidelberg, 2003. [Xekalaki, 1983alE. Xekalaki. Infinite divisibility, completeness and regression properties of the univariate generalized waring distribution. Annals of the Institute of Statistical Mathematics, 35279-289, 1983. [Xekalaki, 1983blE. Xekalaki. The univariate generalized waring distribution in relation to accident theory: Proneness, spells or contagion? Biornetrics, 39:887895, 1983.
Estimation of simple characteristics of samples from skewed and heavy-tailed distributions Zdenek Fabi&n Institute of Computer Science Academy of Sciences of the Czech Republic Pod vodarenskou veii 2 Prague 18200 Czech Republic (e-mail: zdenekQcs.cas .cz) Abstract. Abstract. We present new characteristics of the central tendency and dispersion of data samples. They are constructed from estimates of parameters of underlying distributions and make possible an easy comparison of results obtained under different assumptions. Keywords: scalar inference function, generalized moment method.
1
Introduction
Johnson score - a scalar inference function - was introduced by [1,2] for a large class of continuous probability distributions. It was shown that the Johnson score moments exist under mild regularity conditions even in cases of distributions without mean and variance. The first moment describes the central tendency of the distribution and the reciprocal value of the second moment the dispersion of the values around the central point. It seems that whereas the mean m = J z d F ( z ) and variance g2 = z 2 d F ( z )-m2 compare the properties of distribution F with the standard (with the the normal distribution), the new characteristics have an ability t o compare distributions within parametric families even when distributions are skewed and/or heavytailed. Usually, having a n idea about the type of the underlying distribution, it is to estimate the parameters. We argued for a slight change of view: there are the sample Johnson mean and sample Johnson variance, which are to be estimated as characteristics of data samples taken from the distribution under consideration. They make possible t o compare results of estimation for various assumed distribution families parametrized by arbitrary ways.
2
Johnson score
Let us define the basic concept. Definition 1. Let F be distribution with support X = ( a ,b) C_ R and density f continuously differentiable according t o the variable. Let mapping 43
44
Recent Advances in Stochastic Modeling and Data Analysis
77 : X
+
R be defined by f
77b)=
x log(x - a ) (x - a ) log iogjb - .j
if ( a , b ) = R if-ca
< a < b < 00 - m = a < b < 00,
if -cc if
let function T ( z ) be given by
and the solution x* of equation T ( x )= 0
(3)
be unique. Function S(x) = q’(x*)T(x)
(4)
will be called a Johnson score of distribution F. Since ~’(x) > 0, x* is the solution of equation
S ( x )= 0
(5)
as well. The philosophy behind Definition 1is the following. Any distribution F with interval support X # R is viewed as a transformed prototype G supported by R, that is, F(x) = G(q(x)), x E
X.
(6)
Mapping q given by (1) is the Johnson transformation [3] adapted for arbitrary interval support. Denoting by g the density of G, the density of the transformed prototype (6) is f ( x ) = 9(77(2))77‘(x),
xE
x
(7)
where ~ ’ ( 2is)the Jacobian of the transformation. Let Q be the score function of G.
While the score function can be taken as a suitable inference function of prototypes, the transformed score function of the prototype,
T ( x )= Q ( v ( x ) ) ,
(9)
was found to be a relevant inference function of (6. It is termed by [4] a core function of F . Formula (2) follows from (9) by using (8) and (7) and shows
Estimation f r o m Skewed & Heavy- Tailed Distributions
45
that the core function can be determined without reference to the prototype by a special type of differentiating of the density according to the variable. Let G(y-p) be a prototype with location parameter p E R (expressing its central tendency). Consider distribution which is the transformed G(y - p ) on ( a ,b ) # R. Set t = 7-l ( P I , (10) (10) is called a Johnson parameter and will be considered even in multiparameter cases to express the central tendency of the transformed distribution, the density and core function of which are f(z; t ) = g ( q ( z )- q ( t ) ) q ' ( z ) and T ( z ;t ) = Q ( q ( z )- q ( t ) ) .It was shown by [4] that
d log f ( z ; t ) = q'(t)T(z; t). at
(11)
-
Johnson score S(z) = q ' ( t ) T ( zt;) of the transformed distribution with Johnson parameter thus equals to the likelihood score for this parameter. However, there are distributions without Johnson parameter and without parameters at all. Johnson score (4) is a generalization of relation (11) for these distributions. The generalization consists of replacing t in term q'(t) by the zero of the core function, which actually means a replacing the location parameter of the prototype G by the mode of G. Let us note that the solution of (3) is unique if G is unimodal; in cases of multimodal distributions we consider that z* is the image of the global maximum of the density. Hence z* is the most important point of the transformed distribution, characterizing its central tendency, and Johnson score can be interpreted as the 'score' for this point, which may or may not be a parameter of the distribution. Densities and Johnson scores of distributions used throughout the paper are given in Table 1.
IDistributionI normal lognormal Weibull Frkchet gamma
X R (0, m)
( 0 ,m) ( 0 ,m) (0,m)
7 f log(z/t)"
f [(Zit)" - 11 f [ l-
(;)"I
beta-prime ( 0 ,m)
Table 1.Densities and Johnson scores of some model distributions. function, B the beta function.
r is the gamma
The use of the modified Johnson transformation (1) in Definition 1 leads to Johnson scores expressed by simple formulas for many commonly used
46
Recent Advances in Stochastic Modeling and Data Analysis
probability distributions. Moreover, it is the only transformation under which the prototype of the lognormal distribution is the normal distribution and the inference function of the uniform distribution is linear.
3
Johnson mean and Johnson variance
The Johnson score moments of distribution F with Johnson score S are defined by
ESk
=
S k ( x )d F ( z ) ,
k = 1 , 2 , ...
(12)
It can be easily seen that ES = 0. From the Cram&-Rao regularity conditions imposed on F it follows that 0 < E S 2 < 03. Definition 2. Let F be distribution regular in the Cram&-Rao sense with Johnson score S. The solution x* of equation S ( x ) = 0 will be called a Johnson mean and value w2 = ( E S 2 ) - l (13) a Johnson variance of F. By (4), (13) turns for distributions with support w2
= (X*)”ET~.
X
=
(0, m) into (14)
Let us discuss Johnson characteristics of distributions from Table 1. The zero of S ( x ) = Q ( x ) of the normal distribution is x* = p . Since EQ2 = l/s2,Johnson mean and Johnson variance are equal to the mean and variance. Parameter t of the lognormal, Weibull and Frkchet distributions is the Johnson parameter. Fig. 1 shows densities and Johnson scores of three particular cases of Weibull distribution with c = 1 (exponential distribution), c = 2 (Rayleigh distribution) and c = 3 (Maxwell distribution). Johnson means of all three distributions are x* = 1, the means (denoted by stars) have similar values, m(1) = x*. Frkhet distribution is a heavy-tailed distribution and its mean m = t r ( 1 - l / c ) and variance o2 = t2[T(1- 2 / c ) - T2(1 - l / c ) ] exist only if c > 1 and c > 2, respectively. Fig. 2 shows the densities and Johnson scores of Frkchet distributions with Johnson mean x* = 1. Neither of the means (the stars) describes the position of the distribution on x-axis. m(1) does not exist. Fig.3 shows densities of Fritchet distributions with various Johnson means. The variability of values around Johnson mean is apparently similar to all four distributions. Actually, they have the same Johnson variance w2= 1. Gamma and beta-prime distributions are examples of distributions without Johnson parameter. By ( 2 ) , the core function of the gamma distribution is T ( x ) = y x - Q so that x* = a/r. Since ET2 = Q, by (14)
Estimation from Skewed & Heavy- Tailed Distributions 47
0 0
2.5
-5‘
1
0
2.5
Fig. 1. Densities (a) and Johnson scores (b) of Weibull distributions with z* = 1, c = 1,2,3. The means m ( c ) are denoted by stars. m ( 1 ) = 1,m(2) = 0.885,m(3) = 0.893.
w2 = (x*)’/a: = a:/y2. Johnson mean and Johnson variance of the gamma
distribution are thus equal to the mean and variance. On the other hand, the mean m = p / ( q - 1) and variance
of the beta-prime distribution exist only if q > 1 and q > 2, respectively. By = (P- P)/(X 11, by (3) x* = P / % ET2 = P d ( P 4 1) and (2)’ Johnson variance is, by (14),
+
+ +
Note that (16) looks like (15) with a ’corrected denominator’. Standard deviation and the square root of Johnson variance of the betaprime distribution with p = q as functions of l / q are compared on Fig. 4. u blows up at l / q = 1/2 whereas ’Johnson deviation’ w is comparable with the simulated average median absolute deviation.
48
Recent Advances in Stochastic Modeling and Data Analysis 1
0 0.5
1
2
0
-6
0.5
1
3
Fig. 2. Densities (a) and Johnson scores (b) of FrBchet distributions with z* = 1, c = 1,1.5,2. The means m ( c ) are denoted by stars. m(1) do not exist, m(1.5) = 2.68, m(2) = 1.77.
4
8
Fig. 3. Densities of Frkchet distributions, t = 1 , 2 , 3 , 4 , w = 1.
4
Estimates
Let X I , ..., X , be random variables i.i.d. according t o Fs, B E 0 , 0 C R" with unknown 6' and 51, ...,x, their observed values. The structure of parameters of various distributions, the consequence of the historical development of mathematical statistics, exhibit a chaotic picture. It is difficult t o compare results of the estimation for distribution families parametrized by different ways.
Estimation f r o m Skewed & Heavy- Tailed Distributions
Fig. 4. Beta-prime distribution. 1 - o,2 - w , 3
-
49
simulated MAD
Both the Johnson mean x* : S ( x ; O ) = 0 and Johnson variance w2 = 1/ES2(6')are functions of 0 and can be constructed from the maximum likelihood estimate of 0. In what follows A N *means 'asymptotically normal'. Since ES2(6')> 0, numbers ?b, = x * ( 6 ' ~ , )and Gk, = w2(6),
s,,
characterize a 'center' and dispersion of the sample. Their asymptotic behavior can be easily established by using the delta method theorem, saying that if 8 is AN(6',a2)and ~ ( 6 ' is ) differentiable a t 6' with ~ ' ( 6 ' # ) 0, ~ ( 8is) AN((p(B),[(p'(6')]2a2) (Corollary to Theorem A, [ 5 ] ,pp.122). Another possibility to estimate z* and w is discussed by [I]. Unlike the usual moments, the sample versions of Johnson score moments cannot be determined without an assumption about the underlying distribution family. On the other hand, by substituting the empirical distribution function into (12), a system of equations
appears t o be an alternative t o the maximum likelihood equations in the whole range of the parameters. The estimates from (17) were shown to be asymptotically normal and, in cases of families with bounded Johnson scores, robust and with relative efficiencies near to one. The first equation of (17) for distributions with Johnson parameter is the maximum likelihood equation for this parameter. New results are obtained for distributions without this parameter. Thus, the first equation (17) for - a ) = 0, from which the estimate of z* = gamma distribution is Cy==l(yxi a/r is 2: = 12-l z i . From the first equation of (17) for the beta-prime distribution, cy=l(qxi - p)/(zi 1) = 0, the estimate of z* = p / q is
cyZl
+
In a simulation study, samples of length 100 were generated consecutively from each distribution listed in rows of Table 3, each with values of 6' giv-
50 Recent Advances in Stochastic Modeling and Data Analysis
ing x*(O) = 1 and w(O) = 1.118. Both z* and w were estimated under the assumption of either distribution listed in headlines of columns in Table 3, where the average values over 5000 samples are summarized. It is apparent that erroneous assumptions often lead to unacceptable estimates (note, however, the similar results obtained under assumptions of the lognormal and beta-prime distributions). By the use of the estimates of the Johnson mean and Johnson variance, it is easy to compare the results of estimation for various assumed distribution families parametrized by arbitrary ways.
0.49 0.64
1.013 1.006 1.016
Weibull
gamma
lognormal beta-prime 1.97 0.997 0.99 0.66
3.41 1.11 1.103 0.55
Frkchet 1.42 2.09 0.931
Table 2. Estimates of Johnson mean and Johnson deviation
Aknowledgements. The work was supported by GA ASCR under grant No. 1ET 400300513. References [I]F a b i h , Z. (2006). Johnson point and Johnson variance. Proc. Prague Stochastics 2006, (eds. HuSkovB and Janiura), Matfyzpress, 354-363. [2] FabiBn, Z. (2007). New measures of central tendency and variability of continuous distributions. To appear in Commun. Stat., TheoryMethods. [3] Johnson, N.L. (1949). Systems of frequency curves generated by methods of translations. Biometrika, 36: 149-176. [4] FabiBn, Z. (2001). Induced cores and their use in robust parametric estimation. Commun. Stat., Theory Methods, 30, 3, 537-556. [5]Serfling, R. J. (1980). Approximation. Theorems of Mathematical Statistics. Wiley, New York.
Estimating robustness and parameter distribution in compartmental models of neurons Noam Peled' and Alon Korngreen' The Mina and Everard Goodman Faculty of Life Sciences and the Leslie and Susan Gonda Multidisciplinary Brain Research Center, Bar-Ilan University, Ramat-Gan, Israel. (e-mail: korngraamai I. biu.ac. i I) Abstract. Our understanding of the input-output function of single neurons has been advanced by biophysically accurate multi-compartmental models. The large number of free parameters in these models requires the use of automated approaches for finding their optimal values in a very multi dimensional parameter-plane. Due to inherent noise in measuring equipment and surroundings determination of the accuracy of the obtained parameters is near impossible. Here we show that finding the parameters distribution via Monte Carlo approach simulations reveals the sensitivity of each parameter to noise. This method allows for the reduction of the complexity of the parameter-plane which in turn enables finding the sensitivity of the model to each parameter and the sensitivity of each parameter to noise. Keywords: Multi-compartmental models, Monte Carlo, Robustness, Noise sensitivity, Global minimum
1
Introduction
The properties and functions of neurons in the CNS have been intensively studied over the past decade [Johnston et al., 2003; Migliore and Shepherd 2002; Stuart et al. 19991 providing exciting new information that revived the discussion about the computational properties of single neurons [Hausser and Mel, 2003; Mel, 2002; Poirazi and Mel, 2001; Polsky et al. 20041. Clearly, understanding complex neuronal computations requires functional models which must take into account several types of neuronal properties, e.g., voltage-gated conductances, conductances distributions etc. This complexity resulted in many parameters in neuronal models being tuned by hand [Rhodes and Llinis, 2001; Schaefer et al., 20031. However, the large number of free parameters causes these models to be ill constrained [Rhodes and Llinas 2001; Schaefer eta]., 20031. Previous studies have shown that parameter search methods can be effective in finding matches between single-neuron models and a target data set [Eichler West, 1996; Vanier and Bower, 19991. Nevertheless, not all parameter search methods perform equally well, and the relative performance of the different methods depends on the model being optimized. Stochastic algorithms such as 51
52
Recent Advances in Stochastic Modeling and Data Analysis
simulated annealing [Kirkpatrick et al., 19831 and genetic algorithms (GAS) [Mitchell, 19961 have been found to be the most successful in constraining single-neuron compartmental models because such algorithms are less greedy and rarely fall into local minima, unlike gradient-descent methods that easily get stuck in a local minimum of the parameter space [Vanier and Bower, 19991. It has been shown that for a search with a large number of free parameters, the GA method may be the most effective [Keren et al., 20051. Additionally, the intrinsic parallelism of this method makes it very well suited for implementation on parallel computers [Eichler West, 1996; Keren et al., 20051. Generally, the model of a neuron can be defined as a function with n parameters describing such properties as the potassium and sodium conductance densities or the capacitance of the plasma membrane. The output of such a function will be the output of electrophysiological measurements; i.e., the current or the voltage of the measured compartment. Over the years the robustness of neuronal models has been discussed extensively [Schneidman et al., 2000; Goldman et al., 2001; Marder and Goaillard, 20061. However, to this day the robustness of neuronal compartmental models remains an extremely fluid concept as no rigorous definition of this property for neuronal models has ever been proposed. In this work we defined robustness as the model sensitivity to small fluctuations of its parameters. We next addressed the problem of estimating parameters in compartmental models in the face of noise. Specifically, the parameter distribution of these models was calculated. These distributions were then used in calculating indices indicating the parameters sensitivity to noise and the sensitivity of the model to small fluctuations in the parameters.
2
Methods
Electrophysiological measurements were used to produce the output of the neuronal model. Our aim is to reveal the model parameters using a form of reverse engineering based on the given output. For that purpose some new variables were defined: DS (Data Set): The model output DStme: The desirable model output, or the output of an electrophysiological measurement a: The parameters values vector
{ a l ,a 2 , . .,a,, ]
a,,,,: The parameters values vector yielding DS,,,,. To solve the reverse engineering problem a metric on the parameters phase-plane was defined, calculating the distance between a given DS to DStme,and yielding zero when the DS is exactly the DS,,, using the x2 criterion,
Estimating Robustness and Parameter Distribution
53
Thus, we obtain an N dimensional function which yields, at the global minimum, zero for atrue.The complexity of the function increases with the number of free parameters resulting with numerous local minima [Achard and De Schutter, 20061. To find the global minimum in such a parameter space stochastic optimization algorithms may be of use [Vanier and Bower, 19991. For this work a genetic optimization algorithm was used [Mitchell, 1996; Keren et al., 20051. Electrophysiological measurements of a neuron vary in output at different points in time due to inherent noise; this noise can be divided into two categories: a) equipment noise, and b) behavioral noise of the neuron itself; the latter can be described as “stochastic” as long as its true nature is unknown. In this work, we dealt only with the equipment noise, which was assumed to be random white Gaussian and with deterministic models, where the nature of the noise inserted to the DS is known. Thus, after defining the noise, synthetic data sets were simulated, by adding white Gaussian noise to DStrue (Fig. 1). Two additional variables were defined to deal with the noise:
DslS : The ith synthetic data set
as : The vector of the parameters values which is the global minimum in the parameter phase plane, based on DSl!. The creation of the synthetic data sets enables us to use Monte Carlo simulation (schematically displayed in figure 1) [Press et al., 20021. By using the set
{a:] it was possible to calculate the parameter distributions.
The distributions generator was the noise, as without adding the noise to the DStrue,the parameter distributions would be built from only one value. The wider the parameter distribution the greater is the parameter sensitivity to the noise. A dispersion measure, the VMR (Variance to Mean Ratio) was applied to compare between several distributions width. The VMR was interpreted as an index which indicates the parameters sensitivity to noise. Furthermore, in order to estimate the model robustness, in each minimum the algorithm converged to, the partial derivative (PDer) of the recording to each parameter was calculated numerically. The PDers indicate how much the distance function is sensitive to small changes in the parameters values. After summing all the PDers of all parameters in all the
as , a set of the average influence of each parameter was calculated ( pder )
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Recent Advances i n Stochastic Modeling and Data Analysis
Figure 1: The fitted parameters from an experiment are used as surrogates for the true parameters. White Gaussian noise is used to simulate many synthetic data sets. Each of these is analyzed to obtain its fitted parameters. The reverse engineering problem, based on a synthetic data set, is not a well defined problem. The synthetic data set is not the true output of a model due to the addition of noise. As a result, no vector exists that satisfies the distance function, and the function never yields 0. The function infimum can be defined as:
inx = dis t (DS,,, ,DS;F) For each minimum the algorithm converged to, there is a difference between each parameter value and its true value. This difference can be defined as the parameter bias:
bias, := alS - alrU, Thus, for a parameter displaying a relatively large PDer value, small fluctuations from als will significantly change the distance function. Therefore, it is possible to suggest that a large PDer value means a high probability of small bias. Finally,
Estimating Robustness and Parameter Distribution
55
in order to scale the models a new variable was defined: Information Index for Model Sensitivity (IIMS):
3
where pder is the ith component of { pder . This equation, based on information theory, calculates the ratio between the PDer distribution and a uniform distribution. When the distribution is uniform, it means that the model is sensitive to all the parameters alike. Therefore, IIMS is a factor describing the model sensitivity to its parameters. 0 I IIMS I 1, when IIMS=l the dependence on all the parameters exhibits the same tendency, and when IIMS=O the model depends on one parameter alone.
Results
3
A model of a soma was constructed with one dendrite connected to the soma and two more connected to the first one using the simulation environment NEURON [Carnevale and Hines, 20061. Each compartment included the basic Hodgkin and Huxley mechanism of potassium and sodium channels [Hodgkin and Huxley, 19521. Each compartment is consisted also of 3 types of parameters (12 parameters for the complete model): El : Leakage reversal potential 0
gK: Maximum sodium channel conductance -
g,
: Maximum potassium channel conductance
The DS,,, was calculated using those parameters values: E, = -54.3mV; S,, = 0 . 1 2 P y ; g, = 0 . 0 3 6 ~ ~ .m A 2 step current of O.lnA cm
amplitude and 90ms duration was injected to the soma. The voltage was recorded from the soma. After analysis of the electrophysiological recordings of L5 pyramidal neurons, the noise amplitude turned to be around +0.5 mV. Figure 2 displays the parameter distributions obtained after calculating more than 100 runs of the Monte Carlo simulation. The VMR and PDer calculated from these distributions are shown in figure. 3. As seen in figure 3A, El, the leakage reversal potential, in each compartment is highly sensitive to noise, in comparison to the other parameters. The sensitivity to noise increases with the compartment
Recent Advances i n Stochastic Modeling and Data Analysis
56
Figure 2: Plot of The 12 parameters distributions in a Hodgkin-Huxley neuron with soma and 3 dendrites.
2
6eL
d
s gW
5
gK
d i SL
dl g N A
dl gK
d2sL
d2gW
d2gK
rUeL
639NA
d? gK
02
Figure 3: Plot of the VMR and PDer of the 12 parameters model. Each set of 3 parameters are in a different compartment (s: soma, d,: dendrite number n). (A) El has the largest VMR in each compartment, and it increases with the compartment distance fi-om the soma. (B) gK has the largest PDer in each compartment, significantly, and it decreases with the compartment distance fi-omthe soma.
Estimating Robustness and Parameter Distribution
57
distance from the soma. Figure 3B shows that g K ,the maximum sodium channel conductance density, is the most influential parameter in each compartment. The influence decreases with the compartment distance from the soma. The IIMS is 0.5646, and as we can see almost all the dependence of the model is on a set of half of the parameters.
4
Discussion
We observed that the El has a relatively large sensitivity to noise. Therefore, when measuring the El in electrophysiological experiments numerous measurements must be considered in order to extract meaningful results. The extreme width of the El spectrum may indicate that it is stochastically conducting in the neuron itself. We also observed that S, is relatively the most influential parameter on the behavior of the neuron. It is most likely that when trying to converge to a global minimum for any model, this parameter will have small bias. In this work we built an infrastructure which calculates the parameters sensitivity to noise and the robustness of the model. This infrastructure is not limited to computation ional neuroscience alone, but can be adapted to any field, after determining the N parameters model, the scalar output and the noise component. Further investigations should take into account the dependence of the distribution
{a;] on the distance function used; e. g., the distribution can change when using other distance function, such as IS1 (chi-square of the inter-spikes intervals), or a more complex one [Keren et al., 20051. Additionally, the distribution is dependent on the characteristics of the noise inserted to the DS; inserting more complex noise, non Gaussian and/or not white may lead to different results.
References [Achard and De Schutter, 2006lP. Achard and E. De Schutter, Complex Parameter Landscape for a Complex Neuron Model, PLoS Computational Biology Vol. 2, No. 7,2006 [Carnevale and Hines, 2006lM. Hines, N. T. Camevale, The neuron book, Cambridge university press, 2006. [Eichler West, 1996lRM. Eichler West, On the Development and Interpretation of Parameter Manifolds for Biophysically Robust Compartments Model of CA3 Hippocampal Neurons (PhD thesis). Minneapolis, MN: University of Minnesota, 1996 [Goldman et al., 2001lMS. Goldman, J. Golowasch, E Marder, LF Abbott, Global Structure, Robustness, and Modulation of Neuronal Models, Journal of Neuroscience, 2 1( 14), 200 1
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[Hodgkin and Huxley, 1952lA. L Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its application to conduction and excitation in nerve.” J Physiol 117(4): 500-44, 1952 [Johnston et al., 2003lD. Johnston, BR. Christie, A. Frick, Gray R, Hoffman DA, Schexnayder LK, Watanabe S, and Yuan LL. Active dendrites, potassium channels and synaptic plasticity. Philos Trans R SOCLond B Biol Sci 358: 667-674, 2003 [Keren et a1.,2005]N. Keren, N. Peled, A. Korngreen, Constraining Compartmental Models Using Multiple Voltage Recordings and Genetic Algorithm, Journal of Neurophysiology 94: 3730-3742,2005 [Kirkpatrick et al., 19831s. Kirkpatrick, CD. Gelatt and MP. Vecchi, Optimization by simulated annealing. Science 220: 671480, 1983. [Marder and Goaillard, 2006lE. Marder and JM. Goaillard, Variability, compensation and homeostasis in neuron and network function, Nature Reviews Neuroscience 7, 563574,2006 [Mel, 2002]BW. Mel, What the synapse tells the neuron. Science 295: 1845-1846,2002. [Mitchell, 1996lM. Mitchell, An Introduction to Genetic Algorithms. Cambridge, MA: MIT Press, 1996 [Migliore and Shepherd 2002lM. Migliore and GM. Shepherd, Emerging rules for the distributions of active dendritic conductances. Nat Rev Neurosci 3: 362-370, 2002. [Poirazi and Mel, 2001lP. Poirazi and BW. Mel, Impact of active dendrites and structural plasticity on the memory capacity of neural tissue. Neuron 29: 779-796,2001. [Polsky et al. 2004lA. Polsky, BW. Mel, and J. Schiller, Computational subunits in thin dendrites of pyramidal cells. Nat Neurosci 7: 621427,2004. [Rhodes and Llina’s, 20011 PA. Rhodes and RR Llina’s, Apical tuft input efficacy in layer 5 pyramidal cells from rat visual cortex. J Physiol 536: 167-187,2001. [Schaefer et al., 2003lAT. Schaefer, ME. Larkum, B. Sakmann, and A. Roth, Coincidence detection in pyramidal neurons is tuned by their dendritic branching pattern. J Neurophysiol89: 3 143-3154,2003. [Schneidman et al., 2000lE Schneidman, I Segev, N Tishby, Information capacity and robustness of stochastic neuron models, MIT Press, 2000 [Stuart et al. 1999lG. Stuart, N. Spruston, and M. Hausser, Dendrites. Oxford, UK: Oxford Univ. Press, 1999. [Vanier and Bower, 1999lM.C. Vanier. and J. M. Bower, A comparative survey of automated parameter-search methods for compartmental neural models, J Comput Neurosci 7(2): 149-71, 1999 [Press et al., 2002lW.H.H Press, S.A. Teukilsky, W.T. Vetteling and B.P. Flannery, Numerical Recipes in C++, Cambridge university press, 2002.
On the Stability of Queues with Negative Arrivals Tewfik Kernane Department of Probability and Statistics Faculty of Mathematics University of Sciences and Technology USTHB BP 32 El-Alia, Algiers, Algeria (e-mail: tkernaneQgmai1.corn) Abstract. We obtain in this paper stability and instability conditions for some single server queues with negative customers of batch or work removals. A negative customer removes a group of customers or an amount of work if present upon its arrival. Under general assumptions of stationary ergodic arrivals of negative customers or services of regular customers, we obtain sufficient stability conditions by strong coupling convergence of the process modeling the dynamics of the system to a stationary ergodic regime. We obtain also instability conditions by convergence in distribution to improper limiting distribution. The systems considered are of batch or amount of work removals, removals of customers from the tail of the queue, removals of customers in service, batches of arrivals and services of regular customers. Keywords: Queues, Negative Customers, Stability.
1
Introduction
This paper deals with the problem of stability for single server queueing systems with regular and negative customers. If a regular customer (or positive) enters in the system he takes place in the waiting zone and waits t o be served according t o a specified service discipline. On the other hand, if a negative arrival occurs it removes one or a batch number of regular customers if present. The notion of negative customers was originated by Gelenbe [Gelenbe, 19911. We consider various models of this type, including batch or work removals, batches of arrivals and services, removals of customers in service or at the tail of the queue. These special models, have been studied for their practical interest in telecommunication and computer systems. They may model virus infections in computer systems, deletion of transactions in databases amongst others. The analysis of stability in queueing systems is of great importance since all other performance characteristics depend on whether or not the system is stable. The stability conditions for some models of negative customers have been derived by [Gelenbe et al., 19911 under i.i.d assumptions on service times and inter-arrivals of negative customers, with single deletion of regular customers. In [Kernane and Aissani, 20061, the sufficient stability 59
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Recent Advances in Stochastic Modeling and Data Analysis
condition for a retrial queue with negative customers was obtained under stationary ergodic service times for regular customers.The main approach used here for deriving the strong coupling convergence to a stationary regime was also previously used for retrial queues in [Altman and Borovkov, 19971 and [Kernane and Aissani, 20061. Harrison and Pitel [Harrison and Pitel, 19961 generalized the theory on negative arrivals to the M/G/l queue. Several authors generalized this concept allowing that a negative arrival removes a batch of customers ([Chao and Pinedo, 1995]), a random amount of work [Boucherie and Boxma, 19951 or indeed all the work in the system (Chao 1995, Jain and Sigman 1996). Other studies considered negative customers that accumulate at the nodes of networks but do not need services from the server ([Boucherie and Van Dijk, 19941, [Henderson, 19931, and Henderson e t al. (1994). Another recent interesting generalization is that of Zhu and Zhang (2004) who considered an M/G/l queue with positive and negative customers that cancel each other out and server provides service to either a positive or a negative customer. Our contribution in this paper is based on constructing explicit stochastic recursive sequences which model the dynamics of the considered models and allowed us to obtain explicit stability and instability conditions under general assumptions of stationary ergodic (without independence assumption) service times or inter-arrivals of negative or positive customers. We use the notion of strong coupling convergence to the stationary ergodic regime which implies the coupling convergence and then the convergence in total variation. In the subsequent section, we study the stability and instability of models with removal of customers in service under general stationary and ergodic inter-arrivals of negative customers or service times of regular customers. We also consider that the arrivals are in groups of random sizes and service is also provided for random batches. In section 3, we consider removals of customers at the tail of the queue and batches of arrivals and services under general stationary and ergodic service times or inter-arrival times of negative customers. We also generalize models of negative customers to the case where the negative arrivals not only removes regular ones but breaks also the server. We consider also a model of retrial queue with batches of arrivals, services and removals. In section 4, we consider a system with work removal and batches of arrivals and services.
2 2.1
Removal of Customers in Service Stationary ergodic arrivals of negative customers
Consider a single server queue at which the occurrences of arrivals of regular customers follow a Poisson process with rate A+, and at each arrival epoch a random batch of size ai,with ai an i.i.d sequence with mean ti, enters to the system. Denote by T? the inter-arrival times of regular customers. The waiting zone is of infinite capacity and we assume work conserving disciplines
On the Stability of Queues with Negative Arrivals 61 such as FIFO, LIFO or random access to the service. In this section, we consider the case when a negative arrival removes the customers (if any) in service (RCS). A negative arrival has no effect on an idle system. Negative custoniers arrive at times t,; n = 0,1, ..., and denote by T; = tn+l - t , the inter-arrival times of negative customers. We assume that T; is a stationary (in the strict sense) and ergodic sequence (without independence assumption). Ergodicity means essentially that time averages converge to constants almost surely. Service of regular customers is provided in batches of random size b j , with bj an i.i.d sequence with mean 6, and the time Sj it requires to serve them is exponentially distributed with rate p+. The negative arrival removes the batch bj which is already in service. Denote by N,+ ( t ) (respectively N,+(t))the counting Poisson process with parameter A+ (resp. p + ) which counts the number of occurrences of arriving regular customers (resp. services) during a time interval (0, t ) . We assume that the input flows of customers, sizes of batches and service times are mutually independent. Let Q(t) be the number of customers present in the system at time t. We consider the process Qn embedded immediately after time t , (i.e., Qn = &(in+)). The process Qn can be represented as a stochastic recursive sequence (SRS) as follows:
where (z)+ = maz(0, z). Denote by
i=l
i=l
r
Put VO= 0 and Vn = CY=,E-i. The following proposition gives a stability condition by strong coupling convergence to a unique stationary ergodic regime and instability condition by convergence in distribution to improper limiting sequence.
z,
Proposition 1. If (A% - p+6)ErC < t h e n the process Q , i s strong coupling convergent t o a unique stationary ergodic regime Q, such that QO = SuPn20 Vn. If (A% - p+z)E~; > 6,then the process Qn converges in distribution t o a n improper limiting sequence.
Proof. We have from Wald's equation and the memoryless property of the Poisson process EEn = A+ZErF - p'bEr1 - 6. (3) Since r; is a stationary and ergodic sequence then Sn, which is generated by T; (and other i.i.d sequences), is also a stationary ergodic sequence (see
Recent Advances in Stochastic Modeling and Data Analysis
62
[Kernane and Aissani, 20061). If condition (A% - pL+6)E71 < 6 is satisfied then EEn < 0. From this and Example 11.1 of [Borovkov, 19981, there exists a stationary sequence of renovating events with positive probability for Qn. Using Theorem 11.4 of [Borovkov, 19981,-the SRS Qn is strong coupling convergent to a unique stationary sequence Q, = UnQ0, where is measurable with respect to the a-algebra generated by 7; and U is the shift transformation of random variables generated by 7; (i.e., 7i+1 = V7;). Since T; is ergodic then is also ergodic from the fact that it is an U shifted sequence. From Loynes’ scheme [Loynes, 19621 (for more details see [Gyorfi and Morvai, 20021) - it follows that the SRS is coupling convergent to the sequence Qn where QO= sup,>o V,. Since the strong coupling convergence implies the coupling then the result follows. For the instability condition, if it holds that (A% - p+6)E~; > 6 then E& > 0. It is well known that for SRS of the form Qn+l = ( Q n En)+, the condition E& > 0 implies that the SRS converges in distribution to an improper limiting sequence (see Theorem 1.7 in [Borovkov, 19761).
00
an
-
+
Remark 1. In the case of an i.i.d service times (general distribution B ) and
N ( t ) the counting process of a non-delayed renewal process having cycle lengths i.i.d.-B, then we obtain a more general result, that is for the condition of stability X%E7, - ~ N ( T ;<) 6. 2.2
Stationary ergodic service times
We assume now that the service times S, are stationary and ergodic and arrivals of negative customers occur according to a Poisson process with rate A-. Since the service times Sn are stationary then they have the same distribution function B ( t ) and assume it has a Laplace-Stieljes transform (LST) B * ( s )= B ( t )exp(-st)dt. If it possesses a density b(t) then we denote by p*(s) its corresponding LST. Define s, the instant at which the ( n- 1)st service time ends. Service times, input flows of customers, size of batches are assumed mutually independent. We consider the process Q, embedded immediately after time s, (i.e., Qn = Q(sn+)). The process Qn satisfies the following recursion:
som
In this case EEn = X+ZE(min(S1, 71))- 5. Since tributed with rate A- then we have
E(min(S1,Tc))=
71 is exponentially
(1 - B(s))exp(-sX-)ds =
(1 - A-B*(X-))
x-
dis(5)
or with the density we have E(Tnin(s1,71))= (1 - P*(X-))/X-. We have the following proposition for which the proof is omitted since similar to that of Proposition 1.
O n the Stability of Queues with Negative Arrivals 63 Proposition 2. If A%(l - P*(A-)) < A-b, t h e n the process Q , i s strong coupling convergent t o a unique stationary ergodic regime Q, such that QO = SuPn>oV,. If A%(l - p*(A-)) > A-6, t h e n the process Qn converges in distribution to a n improper limiting sequence. I
Remark 2. By assuming an i.i.d sequence for inter-arrival times of negative customers with exponential distribution with rate A- = ( l / E ~ c and ) single arrivals and services, hence ii = 1 and 6 = 1, we obtain in Proposition 1 the well known stability condition for M/M/l queue with negative customers (see [Harrison and Pitel, 19931): (A+
-
A-)(l/p+)
< 1.
(6)
In Proposition 2, if we assume an i.i.d sequence of service times with exponenA-) tial distribution with rate p+ = (l/E&), we have p*(A-) = p+/(p+ and condition 6 holds true also for this case.
+
3 Removal of Customers at the Tail of the Queue 3.1
Stationary ergodic arrivals of negative customers
The sequence of inter-arrivals of negative customers {T;} is assumed stationary ergodic and customers are removed from the tail of the queue at the epoch of arrivals t , in batches of random sizes d,, with {d,} an i.i.d sequence with mean 2.We assume that the input flows of customers, size of batches and service times are mutually independent. Let Q , embedded just before the arrival of a negative customer. The representation of Qn as a stochastic recursive sequence (SRS) is given by:
i=l We have the following result:
Proposition 3. If (A% - p+b)E~; < 2,t h e n the process - Qn i s strong-coupling convergent t o a unique stationary ergodic regime Qn such that QO = SuPn>o V n . If (A% - p+6)E~; > 2,t h e n the process Q , converges in distribution t o a n
improper limiting sequence. 3.2
Stationary ergodic service times
We assume now that the service times S, are stationary and ergodic and the inter-arrivals of negative customers are i.i.d exponentially distributed with
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Recent Advances in Stochastic Modeling and Data Analysis
rate A-. The process Qn is embedded immediately after the end of the ( n - 1 ) s t service time. The process Qn satisfies the following recursion:
We have the following proposition:
Proposition 4. If (X+si - X-1)ESl < b, t h e n the process Qn i s strong-coupling convergent t o a unique stationary ergodic regime Qn such that QO = SUPn>o Vn. If - X-E)ES1 > b, t h e n the process Qn converges in distribution t o a n improper limiting sequence.
-
Negative customers breaking the server An interesting generalization in practice is to consider that the negative customer not only eliminates regular customers but also causes a breakdown to the server, which must immediately be repaired. As in computer systems where the entry of a virus causes the elimination of programs and thus requires its treatment by an antivirus, and this time of repair may be random. Assume that the server fails at times. Upon the arrival of a negative customer the server fails and takes immediately a repair time Ri, i = 1 , 2 , .... The sequence of repair times {Ri}is assumed to be stationary and ergodic. We assume that after a repair time the server is as good as new and the service of a customer is cumulative. The SRS modeling the system has now the form: NA+(Sn)
Qn+l=(Qn+
1 i=l
ai-
NA-(sn) NA-(Sn)NA+(Rz) ci+ aj - b n ) i=l j=1 i=l
C
C 1
+
.
(9)
We have the following proposition:
-
Proposition 5. If ( X + t i ( l + X-ER1) - X - E ) E S l < 6, t h e n the process Qn i s strong coupling convergent t o a unique stationary ergodic regime Qn such that QO = V,. If ( A + t i ( l + X-ERl) - A-i?)ESl > 5, t h e n the process Qn converges in distribution t o a n improper limiting sequence. Retrial queueing systems with negative customers We can obtain in a similar manner stability and instability conditions for a special class of queueing systems called retrial queues. We consider arrivals in batches for regular customers and if a group of an arrival finds the server busy then the whole batch joins a group of blocked customers called "orbit". Otherwise, if the batch arrival finds the server idle, one of them takes his service and the others join the orbit. Customers in orbit reapplies at random times to get
O n the Stability of Queues with Negative Arrivals 65 served according to the following versatile policy. The probability of having a retrial during the time interval ( t ,t At],given that j customers were in orbit at time t, is (6(l - 60j) j v ) A t o ( A t ) . This versatile retrial policy, introduced by [Artalejo and Gomez-Corral, 19971, incorporates the classical linear policy and the constant one. If v = 0 we obtain the constant retrial policy studied by [Fayolle, 19951. Denote by u, a sequence of independent random variables uniformly distributed on [0,1] and generating the type of customer that gets the service (external or from the orbit) at the end of the successive service periods. The stochastic recursive sequence modeling the dynamics of the system has the following representation: NA+(Sn) Nx- ( S n ) ci Ci=l Qn+l = (Qn Ci=l ai (a1 - l ) I ( U n I x + + ~ + Q1 ~ ~
+
+
0fQnv A++o+Q,,,
I)+.
+ +
+
bnI(un 5 a1 is the size of the first batch arrival if the external arrival occurs before a retrial from the orbit. We have the following proposition: Proposition 6. 1) If v = 0, 6 > 0 and ( X f E - X-E)ESl < ( A + ~ ~ f ~ e ' ) then th,e proce_ss Qn is stronx coupling convergent to a unique stationary er- V,. godic regime Q, such that QO = sup,>o
If ( X b - X-E)ES1 > then the process Q, converges in distribution to an improper limiting sequence. 2) If v > 0, 0 2 0 and (A% - X-E)ESl < b, then the process- Q, is strong coupling convergent to a unique stationary ergodic regime Q, such that Qo = SUP,>O - Vn. If (A% - X-E)ESl > 6, then the process Q, converges in distribution to a n improper limiting sequence. Remark 3. By considering a system with retrials without negative customers and service is provided in single then we obtain the result of Theorem 7.1 in [Kernane and Kissani, 20061 for a retrial queue with versatile retrial policy and batch arrivals.
4
Work Removals
Consider a single server queue with work removals. The inter-arrival times of regular customers {T:} and service times {S,} are general stationary and ergodic sequences. Let t, the arrival epoch of the nth regular batch of customers. During T:, NA(T;) negative arrivals occur removing t,he with mean E . We assume that i.i.d amounts of work c a , i = 1,...,NA-(T$) the input flows of customers, size of batches and service times are mutually independent. Denote by W ( t )the workload at time t. The following recursion holds for the workload process Wn = W(t,-):
i t1
,
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Recent Advances in Stochastic Modeling and Data Analysis
Denote by En = S , - T ; - ~ Z < ' ~ ' ) cy. We have Etn = ES~-ETF-X-~ET:. Using the same approach as for the precedent propositions we have the following result:
+
< (1 X-E)ET,~,then the process W, is strong coupling convergent to a unique stationary ergodic regime W, such that WO= suPn>o V n . If ES1 > (1 X-E)ET$, then the process W, converges in distribution to an improper limiting sequence. Proposition 7. If ES1
-
-
+
Remark 4. If we assume and i.i.d. sequence of service times with mean 0 a Poisson process with rate A+ = ~/ET; then we obtain the result of stability given in [Boucherie and Boxma, 19951 by X+p < 1 A-2.
+
References [Altman and Borovkov, 1997lE. Altman and A.A. Borovkov. On the stability of retrial queues. Queueing Systems, volume 26, pages 343-363, 1997. [Artalejo and Gomez-Corral, 19971J.R. Artalejo and A. Gomez-Corral. Steady state solution of a single-server queue with linear repeated requests. J. Appl. Prob., volume 34, pages 223-233, 1997. [Borovkov, 1976lA.A. Borovkov. Stochastic Processes in Queueing Theory. John Wiley and Sons, 1976. [Borovkov, 1998lA.A. Borovkov. Ergodicity and Stability of Stochastic Processes. John Wiley and Sons, 1998. [Boucherie and Boxma, 1995lR.J. Boucherie and O.J. Boxma. The workload in the M/G/l queue with work removal. Probab. Engineering and Informational Sci., volume 10, pages 261-277, 1995. [Boucherie and Van Dijk, 1994lR.J. Boucherie and N.M. Van Dijk. Local balance in queueing networks with positive and negative customers. Ann. Operat. Res., volume 48, pages 463-492, 1994. [Chao, 1995lX. Chm. A queueing network model with catastrophes and product form solution. Operations Research Letters, volume 18, pages 75-79, 1995. [Chao and Pinedo, 1995lX. Chao and M. Pinedo. Networks of queues with batch services, signals and product form solutions. Oper. Res. Letters, volume 17, pages 237-242, 1995. [Fayolle, 1995lG. Fayolle. A simple telephone exchange with delayed feedbacks. Teletrafic Analysis and Computer Performance Evaluation (0.J. Boxma, J. W. Cohen, and H. C. Tijms, eds.), Elsevier Science, Amsterdam, pages 75-79, 1986. [Gelenbe et al., 19911E. Gelenbe, P. Glynn, and K. Sigman, Queues with negative arrivals. J. Appl. Prob., volume 28, pages 245-250, 1991. [Gelenbe, 19911E. Gelenbe, Queueing networks with negative and positive customers and product form solution. J. Appl. Prob., volume 28, pages 656463, 1991.
On the Stability of Queues with Negative Arrivals 67 [Gyorfi and Morvai, 2002lL. Gyorfi and G. Morvai. Queueing for ergodic arrivals and services. In I. Berkes,(ed.) et al., Limit theorems in probability and statistics. Fourth Hungarian colloquium on limit theorems in proba. and Stat, Balatonlelle, Hungary, 1999. volume 11. Budapest: Janos Bolyai Mathematical Society pages 127-141, 2002. [Harrison and Pitel, 1993lP.G. Harrison and E. Pitel. Sojourn times in single-server queues with negative customers. J. Appl. Prob., volume 30, pages 943-963, 1993. [Harrison and Pitel, 1996lP.G. Harrison and E. Pitel. The M/G/l queue with negative customers. Adv. Appl. Prob., volume 28, pages 540-566, 1996. [Henderson, 1993lW. Henderson. Queueing networks with negative customers and negative queue length. J. Appl. Prob., volume 30, pages 931-942, 1993. [Henderson et al., 1996lW. Henderson, B.S. Northcote and P.G. Taylor. Geometric equilibrium distributions for queues with interactive batch departures. Ann. Operat. Res., volume 48, pages 493-511, 1994. [Jain and Sigman, 1996lG. Jain and K. Sigman. A Pollaczek-Khintchine formula for M/G/1 queues with disasters. J. Appl. Prob., volume 33, pages 1191-1200, 1996. [Kernane and Aissani, 2006lT. Kernane and A. Aissani. Stability of Retrial Queues with Versatile Retrial Policy. Journal of Applied Mathematics and Stochastic Analysis, 2006:pages(16), 2006. [Loynes, 1962lR.M. Loynes. The Stability of a Queue with non-independent interarrival and Service Times. Proc. Cambridge Philos. SOC.,58:497-520, 1962. [Zhu and Zhang, 2004lY. Zhu and Z.G. Zhang. M/GI/l queues with services of both positive and negative customers. J. Appl. Prob., volume 33, pages 11911200, 1996.
Random Multivariate Mult imodal Distributions George Kouvaras' and George Kokolakis2 National Technical Uiiiversity of Athens Department of Mathematics Zografou Campus 15780 Athens, Greece (email: gkouvhath. ntua. gr) National Technical University of Athens Department of Mathematics Zografou Campus 15780 Athens, Greece (email: Kokolakishath. ntua. gr)
Abstract. Bayesian nonparametric inference for unimodal and multimodal random probability measures on a finite dimensional Euclidean space is examined. After a short discussion on several concepts of multivariate unimodality, we introduce and study a new class of nonparametric prior distributions on the subspace of random multivariate multimodal distributions. This class in a way generalizes the very restrictive class of random unimodal distributions. A flexible constructional approach is developed using a variant of Khinchin's representation theorem for unimodal distributions. Results using our approach in a bivariate setting with a random draw from a Dirichlet process are presented. Keywords: Convexity, Dirichlet process, Unimodality-Multimodality, Polya trees, Random probability measures.
1 Introduction Much of nonparametric Bayesian inference has proceeded by modelling the unknown cumulative distribution function (c.d.f.) as a stochastic process. In a fundamental paper, [Ferguson, 19731, a random process, called the Dirichlet process, was defined as a distibution on ( P , S ) ,where P is the collection of all probability measures on a measurable space (X,.F), endowed with a calgebra S . The major drawback of a Dirichlet process is that it selects discrete distributions with probability one [Ferguson, 19731 and [Blackwell, 19731. Several different classes of nonparametric priors, which all contain the Dirichlet process as a particular case, have been proposed. It seems worth mentioning, among others, the mixture of Dirichlet processes [Antoniak, 19741, which is a Dirichlet process where the base measure is itself random and the mixture of Dirichlet process prior [Lo, 19841, which is a convolution of a Dirichlet process with an appropriate kernel. After the work of previous authors, the study of absolutely continuous random probability measures has 68
Random Multivariate Multimodal Distributions
69
become a very active area of research, touching on both analytical and simulation based approaches, cf. Polya Trees [Lavine, 19921, [Lavine, 19941 and [Kokolakis and Dellaportas, 19961, Dirichlet diffussion trees [Neal, 20011 and Levy-driven Processes [Nieto-Barajas and Walker, 20051. For reviews of nonparametric priors the reader is referred to [Walker et al., 19991, [Hjort, 20031, [Dey, 19981, [Ghosh, 20031 and the references therein. In this paper, we present a Bayesian nonparametric inference for unimodal and multimodal random probability measures on a finite dimensional Euclidean space that have finite expected number of modes. As a consequence, we get a random probability measure that admits a derivative almost everywhere in IRd. The paper is organized as follows. Section 2 has the essential theoretical background on multivariate unimodality to implement our methodology. In section 3, a detailed description of partial convexification procedure is provided. Random bivariate multimodal probability measures are constructed and possible modifications and extensions are discussed in section 4.
2
Univariate and multivariate unimodality
An important property of a distribution is unimodality. A univariate c.d.f. F is said to be unimodal with mode (or vertex) at m, if F is convex on (-m, m) and concave on ( m ,GO). We make use of unimodality to get absolutely continuous distribution functions. 2.1
Unimodality on R
For univariate distributions there is a well known representation theorem due to Khinchin (see [Feller, 19711, p.158) that refers to the classical univariate unimodality.
Theorem 1. A real valued random variable X has a unimodal density at 0 i f and only i f it is a product of two independent random variables U and Y , with U unaformly distributed on ( 0 , l ) and Y having an arbitrary distribution. This can be expressed in the following equivalent form cf. [Shepp, 19621, [Brunner, 19921 and (Kokolakis and Kouvaras, 20071.
Theorem 2. The c.d.f. F is convex on the negative real line and concave on the positive, if and only i f there exists a distribution function G on R such that F admits the representation:
for all x points of continuity of G .
70
2.2
Recent Adva.nces in Stochastic Modeling and Data Analysis
Unimodality on Rd
For multivariate distributions, however, there are several different ways that unimodality is defined. Among the main types of multivariate unimodality there are the following: the beta unimodality, which is generated by the Beta distribution, Beta(m,u ) , instead of Uniform distribution on the interval (0, l), and contains the classical univariate unimodality and some of the existing multivariate notions of unimodality as special cases (star unimodality, u-unimodality), the linear unimodality, which is characterized by the unimodality of the distribution of any linear combination of the components of a random vector and the strong unimodality, which is defined as a convolution of unimodal distributions. An extended study of different types of unimodality and their useful consequences can be found in [Dharmadhikari and Kumar, 19881 and [Bertin et al., 19971. In what follows we focus our attention on the Khinchin’s classical unimodality extended to the multivariate case. For the sake of simplicity we restrict ourselves to the bivariate case. Results for higher dimensions can be easily derived. According to the classical unimodality [Shepp, 19621 we have: Theorem 3. The c.d.f. F is unimodal at 0 i f and only i f there is a random vector ( X l , X 2 ) with c.d.f. F , such that ( X l ,X 2 ) = ( K U 1 ,Y2U2)l
(2)
where (Y1,y2) and ( U l ,U2) are independent random vectors, (U1,U2) is uniformly distributed on the unit square and (Y1,Yz) having an arbitrary c.d.5 ti.
Equivalent to Theorem 3 is the following.
Theorem 4. The c.d.f. F is unimodal at 0 i f and only i f for all points of continuity of G ,
( 2 1 , ~ ~ )
F ( z i , 2 2 ) = ~ 1 F ’ ~ ( 2 1 , 2 2+ ) z2FZ2(51,z2)- z i ~ f ( z i , z 2+)G(21,22), (3) where G is an arbitrary c.d.f., with subscripts in (3) denoting partial derivatives. According to the above procedure, i.e. by the component wise multiplication of two independent random vectors (Y1,Y2) and (U1,U Z ) ,where the latter is uniformly distributed on the unit square, we always get a c.d.f. F with a single mode at zero, no matter what the distribution G, we start with, is. To overcome the limitation of getting always a single mode at zero we propose the following “partial convexification” procedure.
3 Univariat e and multivariate partial convexification Partial convexification procedure of a c.d.f. G relies on using U ( a ,1) distributions, with 0 < a < 1, instead of U ( 0 , l ) [Kokolakis and Kouvaras, 20071.
Random Multivariate Multimodal Distributions
71
The parameter (Y can be fixed, or random with a prior distribution p ( ~ ) , on the interval (0, 1). According to this, we obtain a prior distribution on the subspace of multimodal c.d.f.’s. The expected number of modes of F increases from one, when (Y = 0, to infinity, when a = 1, having a finite number of modes when 0 < LY < 1. This means that when 0 < LY < 1, the c.d.f. F ( x ) alternates between local concavities and local convexities, i.e a “partial convexification” of F is produced. Definition 1. The d-variate c.d.f. F is called partially convexified if there exists a random vector X = (XI,.. . ,Xd) with c.d.f. F , such that
(XI,. . . ,Xd) = (YIUl,. . . YdUd),
(4)
where Y = (Yl,.. . ,Y d ) and U = (Ul,. . . ,U d ) are independent vectors, U is uniformly distributed on the rectangle ( ( ~ 11) , x . . . x ( a d , 1) and Y having an arbitrary d-variate c.d.f. G. 3.1
Partial convexification on
IR
Using the partial convexification procedure, Theorem 2 can be expressed [Kokolakis and Kouvaras, 20071 in the following form. Theorem 5. The c.d.f. F is partially convexified i f there exists a distribution function G on R such that F admits the representation:
for all x, z / u , points of continuity of G. 3.2
Partial convexification on
R2
A generalization to the bivariate case using Uniform distributions U ( ( Y l), ~, i = 1,2, with parameters ai fixed in the interval (0, l), is as follows. Theorem 6. The c.d.f. F is partially convexified i f there exists a distribution function G on IR2 such that F admits the representation:
F(xI,~
2= ) X
+
I F ~( ~, 1 , ~ 2 x2FZz ) ( 2 1 , ~-) x 1 ~ 2 f ( x i~,
+ &(xi,~
2 )
2 )(6)
where
for all (x1,x2), (x1/a1,22), (zlrx2/a2) and (xi/ai,x2/a2) points of continuity of G.
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Recent Advances in Stochastic Modeling and Data Analysis
Proof. We have: F(Xl,X2) =
P[X1 I
X l , X 2
i x 2 1 = P[UlYl 5 X l , U Z Y 2 5 5 2 1
For xi 2 0,
for x;
<0
(i = 1,2)
and Ho(.) is degenerate at zero. In the sequel we will only refer to the first quadrant 11 = {(x1,x2) : X I 2 O , x 2 2 0). Thus in 1 1 , the expression (8) can be written as follows:
F(Xl,X2)
=
J
(-c=711
G( dYl, dY2) x (--w,=21 2 2 -02y2
21 -
21
WYl
- f f l Y l x2
- 02Y2
G(dY1, d Y 2 ) .
(9)
With fixed yi # 0, the functions Hy,(.)(i = 1 , 2 ) are bounded with bounded left and right derivatives a.e. in I l . Applying the bounded convergence
Random Multivariate Multimodal Distributions
73
theorem we conclude that the c.d.f. F is differentiable a.e. in I1, with respect to the Lebesgue measure, and their 1st and 2nd order mixed derivatives, wherever they exist, are as follows:
and
Introducing the above results in (9) we get finally:
where
We may notice that when the points ( 2 1 , 2 2 ) , ( 2 1 / a 1 , 2 2 ) , ( 2 1 , 2 2 / a 2 ) and ( z l / a l , ~ / a 2 ) are all continuity points of G, then Q(z1,zz) takes the form:
Expressions (10) and (11) hold to the others quadrants as well. When the origin (0,O) is a continuity point of G then f is a density.
74 Recent Advances in Stochastic Modeling and Data Analysis ._
4
om bivariate multimo~alprobability ~ e a s u r e s
In our Bayesian model specification we assume the following:
-
(Y1,Y2) G, where G is a random c.d.f. produced by a Dirichlet process, using the tree structure presented in [Kokolalcis and Kouvaras, 20071. ( U l , U2) is uniformly distributed on the rectangle ( a l ,1) x (az,1) with a, fixed on the interval (0, 1). (Yl ,Y2) and (U1, U2) arc independent.
By definition, the c.d.f. F of the random vector ( X 1 , X z ) = (YlUl, Y2U2) will be random partially convcxified c.d.f. and thus we take a prior distribution on the subspace of bivariate multimodal c.d.f.'s. For demonstration purposes, we consider the generation of a random c.d.f G from a Dirichlet process, DP[c/3(.)],where the concentration parameter is c = 2 and the base distribution /?(-) on Rzis a mixture of two bivariatc normal distributions. Specifically, a(y) = wlN(yl1-11,CI) w2N(yylp2,62) where,
+
1-11
=
(;),
1-12
=
(;;),
.El=
(; ;), c; ( ,) =
16 4 4
with weights wl = 0.4 = 1 - w2. The sample sizes have all been taken equal to 250. The partial convexification procedure has been applied with the parameter a = a1 = a2 = 0.80.
(a) Before Partial Convexification.
(b) After Partial Convexification.
Fig. 1. Construction of a Random Bivariate Muitimodal Distribution Using Partially Convexified Procedure.
In Figure l(a), a random c.d.f. G from a Dirichlet process is presented. In Figure l(b), partially convexified version is presented. We may notice the partially convcxification procedure results in some overestimation of the posterior distributions. Various generalizations of the present result for higher dimensions and correlation structures arc possible. Such models can be useful alternatives to the standard Bayesian mixtures models for multivariate multimodal distributions.
Random Multivariate Multimodal Distributions
75
References [Antoniak, 19741Ch. Antoniak. Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist., 2:1152-1174, 1974. [Bertin et al., 1997lE.M-J. Bertin, I. Cuculescu, and R. Theodorescu. Unimodality of Probability Measures. Kluwer Academic Publishers, 1997. [Blackwell, 19731D. Blackwell. Discreteness of Ferguson selections. Annals of Statistics, 1~356-358, 1973. [Brunner, 1992lL.J. Brunner. Bayesian nonparametric methods for data from a unimodal density. Statistics and Probability Letters, 14:195-199, 1992. [Dey, 1998]Muller P. Sinha D. Dey, D. Practical nonparametric and semiparametric Bayesian Statistics. New York: Springer-Verlag, Inc, 1998. [Dharmadhikari and Kumar, 19881s. Dharmadhikari and J-D. Kumar. Unimodality, Convexity and Applications. Academic Press, 1988. [Feller, 19711W. Feller. A n Introduction to Probability Theory and its Applications, Volume 2. Wiley, New York, 2nd edition, 1971. [Ferguson, 1973lTh. S. Ferguson. A Bayesian analysis of some nonparametric p r o b lems. Annals of Statistics, 1:209-230, 1973. [Ghosh, 2003]Ramamoorthi R. Ghosh, J. Bayesian Nonpammetrics. New York: Springer-Verlag, Inc, 2003. [Hjort, 2003lN. Hjort. Topics in nonparametric Bayesian statistics. In N. Hjort P. Green and S. Richardson, editors, Highly structured stochastic systems. Oxford: Oxford University Press, 2003. [Kokolakis and Dellaportas, 1996lG. Kokolakis and P. Dellaportas. Hierarchical modelling for classifying binary data. In Bayesian Statistics 5, pages 647452, London, 1996. Oxford University Press. [Kokolakis and Kouvaras, 2007lG. Kokolakis and G. Kouvaras. On the multimodality of random probability measures. Bayesian Analysis, 2:213-220, 2007. [Lavine, 19921M. Lavine. Some aspects of Polya tree distributions for statistical modelling. Annals of Statistics, 20:1222-1235, 1992. [Lavine, 1994lM. Lavine. More aspects of Polya tree distributions for statistical modelling. Annals of Statistics, 22:1161-1176, 1994. [Lo, 1984lA.Y. Lo. On a class of nonparametric estimates: I. Density estimates. Annals of Statistics, 12:351-357, 1984. [Neal, 2001lR. Seal. Defining priors for dist,ributionsusing Dirichlet diffussion trees. Technical report, Department of Statistics, University of Toronto, March 2001. [Nieto-Barajas and Walker, 2005lL.E. Nieto-Barajas and S.G. Walker. A semiparametric Bayesian analysis of survival data based on Levy-driven processes. Lifetime Data Analysis, 11:529-543, 2005. [Shepp, 1962lL.A. Shepp. Symmetric random walk. Transactions of the American Mathematical Society, 104:144-153, 1962. [Walker et al., 199915. Walker, P. Damien, P.W. Laud, and A.F.M. Smith. Bayesian nonparametric inference for random distributions and related functions. J. Roy. Statist. SOC.,Ser. B, 61:485-527, 1999.
A system maintained by imperfect and perfect repairs under phase-type-distributions Delia Montoro-Cazorla', Rafael Perez-0con2, and Ma Carmen Segovia3
' Departamento Estadistica e 1.0 Universidad de Ja6n Jab, Spain (e-mail: dmontoro@,uiaen.es) Departamento de Estadistica e 1.0 Universidad de Granada 18071 Granada, Spain (e-mail: [email protected]) Departamento de Estadistica e 1.0 Universidad de Granada 18071 Granada, Spain (e-mail: msegovia@,ugr.es)
Abstract. We study a system maintained by means of imperfect repairs before a perfect repair. This model has been previously studied by Biswas and Sarkar (2000), which calculated the availability of the system when the operational and repair times were exponential. We extend the model considering these times phase-type distributed. Performance measures are calculated in stationary regime. Keywords: Phase-type distribution,repairs, maintenance.
1 Introduction In certain practical situations it results convenient to maintain operational the system the most time, even its conditions are not the optimal. Mostly for reasons of time and costs, the repairs are frequently imperfect; up to a perfect repair is necessary. In this paper we consider a system that will be perfectly repaired after a certain number of imperfect repairs. The successive operational times after repairs are stochastically decreasing, and the successive repair times are stochastically increasing. This model is studied by Biswas and Sarkar (2000) assuming general distributions for the operational and repair times, but certainly, if the distributions were not exponentially distributed, it would be intractable the application of the main result. We consider phase-type distributions, which can approximate any continuous distributions defined on the positive real line. Moreover, introducing these distributions, the model will be governed by a 76
A S y s t e m Maintained by Imperfect and Perfect Repairs 77
Markov process, and this allows well-structured and tractable expressions. Other related paper are the one of Lim & Lie (2000), Neuts (2000), Perez-Ocon et al. (2004,2004a, 2004b), and references in them. We calculate explicitly the stationary vector of the Markov process that governs the system, and from it, we calculate performance measures of interest, not only the availability, but also the rate of occurrence of failures and the interval reliability.
2 The model In the present paper, we consider the Biswas and Sarkar model, preserving the assumptions and modifying the distribution of the operational and repair times. First, we define the phase-type distributions, which play an important role throughout the paper. Definition I The continuous distribution H(.) on [o,m[ is a phase-type (PHdistribution) with representation (a,T) if it is the distribution of the time until absorption in a Markov process on the states {I ,...,m,m+l) with generator
and initial probability vector (a,am+J,where a is a row m-vector. We assume that the states {I ,..., m) are all transient. Throughout this paper, e denotes a column vector with all components equal to one, the dimension of which is determined by the context. The matrix T of order m is non-singular with negative diagonal entries and non-negative off-diagonal entries and satisfies Te + T o = 0. The distribution H(9 is given by H ( x ) = 1- a exp(Tx)e, x 2 0. It will be denoted that H( 9 follows a PH(a, T ) distribution.
x,
Assumption 1. We denote by the lifetime of the unit after completing (i-I) imperfect repairs. We assume that this random variable Xi follows a phase-type distribution with representation (a(i),T(i)) and with m(i) phases, i=1,2,...k+l. We denote by Y,, i= 1,2,...,k the imperfect repair time of the failure ith, this random variable follows a phase-type distribution with representation (p(i),S(i)) and with
78
Recent Advances in Stochastic Modeling and Data Analysis
n(i) phases. We have: F,(x)=l-a(i) .exp(T(i) -x) .e,Gi(x)=I-P(i) .exp(S(i) .c) -e where e has the appropriate dimension in each case. Assumption 2. We assume that F,,, is stochastically smaller than Fi, i=l,...,k, and G,+,is stochastically larger than Gi, i=l,..,k-I.
Assumption 3. When the system completes k imperfect repairs it is perfectly repaired. The time for completing a perfect repair follows a PH-distribution with representation (y,L), of order n. Assumption 4. All the involved times are independent.
3 Stationary regimen. Realiability measures. 3.1 The generator The methodology we use implies to introduce the structure of states. We will say that the system occupies the macro-state i, i=l,..., k + l , if it is operational and it has completed (i-1) imperfect repairs, it occupies thmacro-state iRif it is being imperfectly repaired after the failure ith, iR=I fi..., kR, and it occupies the macrostate (k+llRwhe it is being perfectly repared.
s(1) Q=
so( M
2 )
... T(k+1) To(k+l)y
A System Maintained by Imperfect and Perfect Repairs
79
The blocks that form this generator can be interpreted in terms of the transition between the macro-states. So, the block that represents the transition I+IR is given by To(l)P(I), since the change occurs when the system undergoes a failure, with absorption vector To(l), and enters repair, with initial vector similarly the generator is calculated.
P( 1). Reasoning
The Markov process that governs the system is positive recurrent, so the system reaches the stationary distribution. We denote by w the stationary probability vector, solution of wQ=O, subjected to the normalization condition we=l. It can be written according to the blocks defined by the macro-states:
Operating, the components of the stationary vector result:
,i = 1,. .., k + 1 i=l
i=l
kcl
k+l
n-(iR)=-,8(i)S-I(i)
with pi=-a(i)T+(i)e, i=I, ...,k+ I , piR=-P(i)S-I(i)e,i=l, ...,k and ,uF+~,R=-YL 6
3.2 Performance measures Now we give the expressions of some well known performance measures, such as the availability, the rate of occurrence of failures (ROCOF) and the reliability in an interval of time. The availability is the probability that the system will be up. It is denoted by A and results:
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Recent Advances in Stochastic Modeling and Data Analysis
since r(i)e =n(l)To(l)pi, i=l,.. ., k+ 1,
A measure of the deterioration is the rate of occurrence of failures (or repairs), that is, the mean number of failures per unit time undergone by the system. This quantity is
In emergency systems, the reliability in an interval is a useful quantity. It represents the probability that the system will be operational during a period of time. Let R(z) be the probability that in the long run the system will not fail in the interval of rank z. For calculating the expression of this quantity, we consider that the system must be in any of the operational macro-states within the interval. Then, we have:
4 Numerical application Let a system be that is imperfectly repaired up to 3 times before its perfect repair, then k=3. The successive operational times after repairing form a geometric processes with a deteriorate factor a=1.25, that is, the operational meantime after an imperfect repair is reduced a 20%. While operational, the system goes through three phases, and the fourth indicates the failure. Then, the operational time after the (i-1)-repair follows a PH-distribution with representation (a,a1-'T),i=l,2,3,4, that is, a(i)=a=( l,O,O), T(i)=a"T. The time of the ith imperfect-repair follows a PH(P,b"S), being b=O. 75, that is, the repair meantime after every repair is
A S y s t e m Maintained by Imperfect and Perfect Repairs 81 enlarged a 25%, that is, p(i)=p=(l,O,O), S(i)=b"'S, i=I,2,3. The perfect repair time follows a PH(y,L), with y=( l,O,O). The matrices T, S, L are:
-3
3
27
-27
-15
15
For this model, we calculate the availability, rate of occurrence of failures, availability in the interval [t,t+5] for t large. Moreover, the measures are calculated for k=O,1,2 and a comparison is established, kl 0
A
I
V
I
R(5)
I 0.8488 I 0.4630 I 0.0093
I,
Table 1. Reliability measures for the Model 'rhc results in the table above illustrate that the highest availability is obtained for the model when k = l .
Acknowledgement: This work was partially supported by Ministerio de Educacion y Ciencia, Spain, under grant MTM2004-03672.
5 Specifying references [Biswas, A. and Sarkar, J.,2000]. Availability of a system maintained throough several imperfect repairs before a replacement or a perfect repair. Statist. Probab. Lett. 50, 105-114. [Lim, T.J. and Lie, C.H., 20001. Analysis of System Reliaiblity with Dependent Repair Modes. IEEE Trans. Reliab., 49(2), 153-162. [Neuts, F. et al. 20001. Repairable models with operating and repair times governed by phase type distributions. Adv. Appl. Probab. 34, 468-479. [PCrez-Ocon, R. and Montoro-Cazorla, D., 20041. Transient Analysis of a Repairable System, Using Phase-Type Distributions and Geometric Processes. IEEE Trans. Reliab., 53(2), 185-192. [PCrez-Ocon, R. and Montoro-Cazorla, D., 20041. A multiple system governed by a quasi-birth-and-death process. Relab. Eng. Syst. Safety, 84, 187-196. [Perez-Ocon, R. and Ruiz-Castro, J.E., 20041. Two models for a repairable twosystem with phase-type sojourn time distributions. Relab. Eng. Syst. Safety, 84, 253260.
Asymptotically robust algorithms for detection and recognition of signals* Veniamin A. Bogdanovich’ and Aleksey G. Vostretso?
’ Saint-Petersburg Electrotechnical University “LETI” Professor Popov str. 5 , St.Petersburg, 197376, Russia (E-mail: [email protected]) * Novosibirsk State Technical University K.Marx av. 20, Novosibirsk, 630092, Russia (E-mail: [email protected])
Abstract. Using the principle of minimax and Bayesian approach the method of synthesis of asymptotically robust (AR) algorithms for detection and recognition of signals with unknown parameters on the background of independent noise and signallike interference is developed. According to this method the asymptotically robust algorithm is sought in the class of algorithms of correlation type with the threshold depending on the observed sampling. For synthesis of a robust algorithm using this class no requirements are imposed on energy parameter of a signal, it is different from the known class of algorithms of M-type. Keywords: nonparametric a priori uncertainty, model of noise distribution, q-point model, finite variance model, principle of minimax, Bayesian approach, asymptotically robust algorithm.
1
Introduction
For the AR algorithm synthesis we use minimax optimality criterion, according to which in some class of expected density of probability distribution (DPD) of noise the maximal asymptotic losses owing to signal omission for fixed level of asymptotic probability of false alarm is minimized. The additive interference is represented as the sum of an independent stationary noise with some scaling parameter LT E (0, a) and a P
signal-like interference 17 = C
with the arbitrary shift parameters
/=I
$, € ( - a , m ) , j = l ,
...)P .
The energy signal ( A / & ) S ( 8 )
A and nonenergy 8 parameters of the being detected are also considered as uncertain. For the parameter A
only the set of its values is established in order to have possibility of construction of AR algorithms which are uniformly asymptotically minimax with respect to this parameter. The parameter 8 is assumed as random but Supported by Russian Foundation for Basic Research (Project No 07-0700070) 82
Asymptotically Robust Algorithms 83
with unknown a priori distribution P ( 8 ) . To overcome uncertainties of the distribution P(8) we use asymptotic Bayesian approach to statistical synthesis. In the case of nonparametric a priori uncertainty the hypothesis Ho about the signal absence and the alternatives H k , k = 1, ...,m , about the presence of some signal are represented, correspondingly, by the families of distributions
{
Pk,n = wn
(xl&e?Y)
= wn[x-ks(e)
I Yl,
wn
(’ I Y) EPo,n,A
‘
(O3.O)
,8 Edk)}
>
(2)
where W is the class of expected DPD of noise, y = (D,4,...,.pP ) is the fixed parameter of the additive interference, d k )is the range of definition of the parameter 8 for the alternatives H,, k = 1, ...,m . The signal S (0) in families (2) is expressed in the form N
(3) j=l
i
1
where S(’) = Sp),...,S r ) ,j
( ’
= 1,...,N
I
is the known orthogonal basis of
the signal, 8 = ( Ql ,. . .,ON ) is a random parameter. In order the energy of the signals
(l/&)e(’),
(l/&)S(’),j
= 1, ..., N
,
i = 1, ...,P , do not tend to infinity with increase of the observed
sampling
size
we
use
the
P, (l/&)lle(i)ll
following
normalizations:
= 1 t’ m = 1,...,P. m
In addition, we assume that the norm ll8ll= 1 t’ 0 E E = IJ
dk).Therefore,
k=l
for all values of the parameter 8 E E the signal
1
(A/&)S@)
II=A
.
(A/&) S (8) has the norm
84
2
Recent Advances in Stochastic Modeling and Data Analysis
Models for a priori uncertainty of noise
Nonparametric a priori uncertainty of noise can be represented by some class W of the densities ~ ( t )of distribution with finite Fisher's information about shift. For the finite Fisher's information the densities w ( t ) are
absolutely
continuous
functions
and
Fisher's
information
m
y$ ( t )w(t)dt , where yw is the logarithmic derivative of the
Z(w) = -m
density w(t) [l]. For synthesis of AR algorithms in the case of independent noise the following models of noise distributions are the most applicable: model with a finite variance, E -pollution model and q -point model. The model with finite variance is represented by the class of DPD of
i
I
m
noise W, = w: J t2w(t)dt= 6,, I( w ) < co , the model of -m
by the class W,
q -point model
-
=
E
-pollution -
(w:w(t)=(l-~)f(t)+~h(t), Z ( f ) < ~ , Z ( h ) < c o and )
-{
I
by the class Wa,q- w: 1 w(t)dt = q, I( w )< co , where a:
all DPDs have zero mean, f is such doubly differentiable DPD that the function -logf(t) is convex, E E (0,1), 6 E (o,co), q E (O,1), a E ( 0 , ~ ) are the parameters of the models. The models W,, W, II Wa,q are mostly investigated. For these models the distributions are found that have minimal Fisher's information about shift and play important role in synthesis of AR algorithm [l]. The distribution density with minimal Fisher's information is called least favourable DPD of noise. For this density we use here and further the notation wo ( t ). In the class W, the least favourable DPD coincides with the density of normal distribution
In the class W, the least favourable DPD is expressed in the form
(1 - ~ ) f ( l)exp[ t k ( t -t
,)I , co
;
(5)
tl I t I t , ;
(l-E)f(t~)exp[-k(l-t2)~ , t*
2
Asymptotically Robust Algorithms 85
where tl
H
t2 are the limits of the range where the inequality holds
l f ’ ( t ) / f ( t ) lI k , k is the parameter related to the value of
E
by the
relationship
If the nominal density f ( t ) = (l/&)
(
exp -t2/2) , then, according to (5)
exp(-t2/2)
, ~ tI( k ,
and (6), the density wo( t )=
In the class Wu,qthe least favourable DPD has the form
wo
where
the
(4 =
I
c cos2 (A/2)
parameters
A
cos2[ A t / ( 2 a ) ] , It1 I a,
and
B
satisfy
the
equations
wo( t )dt = q and the parameter C = cos2(A/2)/( 1+ 2 / B ) .
B = A .tg (A/2), -U
Extending of the models W, and Wu,qis based on the possibility of representation of almost any DPD with zero mean in the form p ( t ) = ( l / c ) w ( t / c ) , where the density w E W, ( W a , q ) .For such representation of DPD it is sufficient to assume that density w ( t )= c p (ct ) m
and to determine the parameter c from the equation
I t 2 p ( t ) d t= e2iT2in
-a2 UO
the case of the model W, or from the equation
I
p ( t ) d t = q in the case of
-UO
the model Wu,q.Owing to arbitrariness of the density p the parameter c for the specified extension of the models W, and WUlqis a priori uncertain. Therefore, the extended models W, and Wu,qare the models of a priori uncertainty of a mixed type and are represented, correspondingly, by the classes of distributions , (r E (O,co), w
E
W, (w E Wu,q)
loss of generality it is possible to assume for the models P, and
that,
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Recent Advances in Stochastic Modeling and Data Analysis
correspondingly, the parameter S = 1 and the parameter a=l . Further the class Watqfor a=l is denoted for conciseness as Wq, and the class Pa,q as Pq.
3
Statistical synthesis of asymptotically robust algorithms
Initial premises. We synthesize AR algorithms for the following initial premises.
1
(
1B. The basis vectors S(’) = S,(’),...,Sk) satisfy the condition max
max
j=l, , N r = l , ,n
of the signal S(0)
J s ~ ’ ~ _ < n=1,2, o < ~...v .
2B. All distributions of the class W have finite Fisher’s information about shift. 3B. The density w(t19) = w,, ( t - 9), where wo(t- 9) is the least favourable DPD in the class W1, where the density w(t I 9) is continuous and positive with respect to the argument t function differentiable with respect to 9 at the point 9 = 0 ; there exists the neighbourhood U of the point 9 = 0 in which the density w(t I 9) = w ( t )[1+9 y ( t )+ a2S( t ,a)] , m
j S2(t,S)W(t)dt
4B.
For
all
densities
wEW
the
aw( r )= M{W, ( t + z)lw(t)} is differentiable with respect to
function t
in the
neighbourhood of the point r = 0 , moreover the value aw ( r = 0) = 0 and the derivative a; (z = 0 ) =M(vA ( t ) l w ( t ) } where ,
d
v, ( t )= --ln
dt
wo ( t ) .
greater than zero, finite and continuous in the neighbourhood of the point z = 0 for all densities w E W . Definition of A R algorithms. Quality of the asymptotic algorithms p = {pfl,n = 1,2,. ..) we estimate by limiting average losses caused by
w ) = lim n(pn,A, w ) and by the limiting omission of the signal n(q,,A, n-tm
’
For sufficiently general conditions the density wo exists, is absolutely continuous and is the only in the class of distributions with finite Fisher’s information [l].
Asymptotically Robust Algorithms
87
probability of false alarm a (p, w) = lim a (pn,w) , where the losses n+oo
KI (pn,A, y) =
5 nk(p,, ,A, y)
and
the
probability of
false
alarm
k=l
operator of averaging,
nk (pn,A, y )
are average losses in the case of
fulfilment of the hypothesis Hk . We remind that level of false alarm probability (level of the algorithm) is the maximal permissible value a of false alarm probability. Let us choose the class Oa = {p : a (p, w ) I a 'd w E W} of all asymptotic algorithms of the level a . Let us define in this class AR algorithm according to minimax optimality criterion. According to this criterion the algorithm pm E @a is called AR algorithm if it satisfies the condition
where A c ( 0 , ~ )is some set of values of the parameter A , on which the algorithm pm is uniformly optimal according to minimax criterion. In the known publications [2, 31 about robustness it is used for construction of AR algorithm, so called, M-type algorithm .
pr
To construct AR algorithm based on the algorithm p:
it is required
that in both cases, when the signal is present and is absent, the following convergences in probability be ensured:
8 k = y g supTn(x,8,y), k = 1,...,m , are
the estimates of the parameter 0 ,
&ZV)
'y,
d
( t )= --In dt
p ( t ) is the function of inertialess transformation of the
1 n sampling x; .~,,(x,e,y>= - S,(8)r~,
&c
P
r=l
the estimates
ek = ygsupZn(x,8,y), k = 1, ...,m , &#)
helping to establish specified level of the algorithm.
p ( a ) is the parameter
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Recent Advances in Stochastic Modeling and Data Analysis
Owing to the fact that for convergences (8) to exist it is required fulfilment of stronger conditions than premises 1B - 5B, therefore, further for construction of AR algorithm we choose A 0 algorithm p; of correlation type with threshold dependent on the observed sampling. Decision functions of the algorithm p; have the form
where the estimates
8k
= argsupT,(x,8, y),
&$’
k
= 1,...,m
, the threshold
dependent on the sampling size is
To construct AR algorithm based on the algorithm with decision functions (9) it is required that in both cases, when the signal is present and is absent, the following convergence in probability take place Cn (x , a , Y ) * P ( a ) J m of
the
W E
w.
(1 1)
Construction of AR algorithm. It can be proved that for large values parameter A the algorithm p t with the parameter
~ ( a= C ) ( a ) / d m is the sought AR algorithm for the class W if premises 1B - 5B are satisfied, thus the following lemma is valid: L e m m a 1 . If premises 1B 3B are satisfied then the algorithm qEo is AR algorithm in the class 6 a ,moreover, optimality of the algorithm with respect to minimax criterion is ensured in this class for all values of the parameter A E (0,a) and any distribution P (0). In the class of all possible asymptotic algorithms the property of the asymptotic optimality of the algorithm qEo with respect to minimax criterion is ensured in limit ~
when the parameter A + co .
Asymptotically Robust Algorithms 89
Note, that when convergence (8) exists the statement of this lemma is also valid for the algorithm . However, optimality of this algorithm
cpg
with respect to minimax criterion is ensured only in the case when the parameter A 2 2np ( a ) / (2 - A ) . For low level of a of the algorithm (large parameter p ( a )) the lower boundary for the parameter too high, that is drawback of the algorithm
A can appear to be
cpg
After applying lemma 1 for constructing AR algorithms for representation of nonparametric a priori uncertainty by the classes of distributions W, , W, and Wqq, we have that AR algorithm is the algorithm Two
with decision functions (9).
For logarithmic derivatives convergence (8) is valid. Therefore, for the classes W, , W, and Wa,s with symmetric DPDs the algorithm can
cpg
also be used as AR algorithm. It can be shown that for specified change of initial premises 1B and 4B the algorithm qEo asymptotically robust in the class &a for symmetric DPD remains robust for the case when no requirement of symmetry is imposed on DPD of noise. Let W, denote the class of symmetric DPDs of noise and W* denote the class of distributions having both symmetric and nonsymmetric DPDs of noise. Let the mean value for the nonsymmetric DPDs be zero. As before, Fisher’s information is considered finite for all DPDs from the class W” . Initial premises 1B and 4B we formulate in the following way: of the signal S(8)
1B‘. The basis vectors S(’) = satisfy the conditions j=I,N r=l,n
4B’. For all densities w E W* the function a, (z) = m { ‘yo ( t + z) (w( t ) } is differentiable with respect to z in the neighbourhood of the point z = 0 , moreover the derivative ah (z
{
I
)) .
= 0) = M ‘y: ( t ) w ( t
The requirement a, (z = 0 ) = 0 is excluded from the initial premises because it can be unfulfilled in the extended class W* of noise distributions. Instead of this requirement the condition S ( j ) = 0 V’
= 1,...,N
is included, owing to the condition it is
n+m
possible to construct AR algorithm in the case of nonsymmetric DPD of noise.
90
Recent Advances in Stochastic Modeling and Data Analysis
It can be proved that the following lemma is valid. L e m m a 2 . Let the conditions of lemma 1 hold for the formulation of premises I B and 4B in the form 1B' and 4B'. Then the algorithm qEo, where wo is the least favourable DPD in the class W,, is
AR algorithm in the set of asymptotic algorithms for the extended class W" . If the condition lim n+m
= O V j = I , ...,N is not satisfied
the algorithms qEo and q g lose the property of asymptotic robustness in the class W* because of appearance of the uncontrolled displacement pw for asymmetric DPD. However, after some modification the given algorithms preserve the property of asymptotic robustness even in this case, but at the expense of some energy losses.
Conclusion To represent nonparametric a priori uncertainty of noise new models are proposed, the models were obtained by extension of the known models: the model of distributions with finite variance and the q-point model. The extension of the models was obtained by introduction of a priori uncertain scaling parameter. Owing to this almost any DPD can be expressed through the distributions of the models by using appropriate scaling parameter. Using the principle of minimax and Bayesian approach the method of synthesis of asymptotically robust algorithms for detection and recognition of signals with unknown parameters on the background of independent noise and signal-like interference is developed.
References Huber P.J. Robust Statistics. John Wiley and Sons. New York. Chichester. Brisbane. Toronto. 1981. Robustness in Statistics/ Edited by R.L. Launer, G.N. Wilkinson. Academic Press. New York. San Francisco. London. 1979. El-Sawy A. H., Vande-Linde V.D. Robust detection of known signal // IEEE Trans. Inform Theory. 1977. Vol. IT-23. pp. 722-727.
Three parameter estimation of the Weibull distribution by order statistics Vaida BartkutC' and Leonidas Sakalauskas' Institute of Mathematics and Informatics, Akademijos 4, Vilnius '(e-mail:[email protected]),
2(e-mail:[email protected]) Abstract. In this paper, we consider the estimation of the three-parameter Weibull
distribution. We construct numerical algorithms to estimate the location, scale and shape parameters by maximal likelihood and simplified analytical methods. Convergence of these algorithms is studied theoretically and by computer modeling. Computer modeling results confirm practical applicability of estimates proposed. Recommendations for implementation of the estimates are discussed, too. Results from simulation studies assessing the performance of our proposed method are included. Key words: order statistics, Weibull distribution, maximum likelihood.
1. Introduction Estimation of three-parameter Weibull distribution [Weibull, 195 11 occurs in many real-life problems. This distribution is an important model especially for reliability and maintainability analysis. The Weibull distribution is one of the extreme-value distributions which is used especially in optimality testing of Markov type optimization algorithms, too [Haan, 1981, Zilinskas and Zhigliavsky, 1991, Bartkute and Sakalauskas, 20041. Because of its useful applications, its parameters need to be evaluated precisely, accurately and efficiently. Three-parameter Weibull estimation has been studied by many researchers but an explicit solution is not known yet. Some questions concerning to the estimation of the location, scale and shape parameters of this distribution by both censored and noncensored samples were considered by several authors [Rockette et al, 1974, Lemon, 1975, Hirose, 1991, etc.]. However, iterative computational methods for the three-parameter Weibull distribution estimation are needed in a most cases [Hirose, 1991, Bartolucci, 19991. In this paper, we present some methods for estimating Weibull parameters, namely, translation, scale and shape parameters using order statistics of noncensored sample. We apply some simplifications which enable us to construct reliable and computationally efficient procedures for estimation. 91
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Recent Advances in Stochastic Modeling and Data Analysis
2. Maximum likelihood and analytical estimation The three-parameter Weibull distribution function is given by:
where c, A and a denote the scale, location and shape parameters, respectively. In this paper we compare analytical and maximum likelihood method for the estimation of these parameters by order statistics of noncensored sample.
2.1 Maximum likelihood estimator This section is concerned with maximum likelihood estimation (MLE) of the three-parameter Weibull distribution. The maximum likelihood methodology is based on large-sample theory and the method might not work well when samples are small or moderate in size. Let m be number of order statistics ~ ( 1 )5 ~ ( 2 2) ... 2 T ( ~ which ) is given
N i.i. random value with distribution function W(.). from the population of Then likelihood function of order statistics can be expressed as follows [Balakrishnan and Cohen, 1990, Zilinskas and Zhigliavsky, 19911:
where w(.)-density function. Now we may write likelihood function for the Weibull distribution which depends on three estimated parameters c, A, a. The general technique for solving for MLE's involves setting partial derivatives of likelihood function logarithm (the derivatives are taken with respect to the unknown parameters) equal to zero and solving the resulting (usually non-linear) equations. Since these equations are nonlinear in the three parameters they can be solved only using nonlinear optimization techniques. However, the standard maximum likelihood method for estimating the parameters of the three-parameter Weibull model can have problems since the regularity conditions are not met; besides, numerical implementation of the method requires of complicated optimization software [Murthy et al., 2004, Blischke, 1974, Zanakis & Kyparisis, 19861. Let propose the modification to simplify the estimation. Hence assume instead asymptotic expansion the next expression is true for some x E [ A , A 61:
+
w(x,a,c, A ) = c(x - A)".
(3)
93
Three Parameter Estimation of the Weibull Distribution
Now let write the likelihood function:
~
l
x-.
m 1 ~..-,qm); ~ A, ~ a))~ =(N-) m). c* , d q m ) -~)a-' -(a-1)dA 1-4qm) - ~ ) a j=lTi)-A
Taking the partial derivative of the log-likelihood function (with respect to each parameter) and setting it equal to 0 we have 6 ,2 estimates:
m
a = m-1
c
2
1n(l+ Pj
(A))
j=l
m
n
C=
where P j
(A)=v(m) V(j ) 1
j=l
A
and
2 , 2 < q(1) is the solution of the equation: 1
j=l
1
(7)
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Recent Advances in Stochastic Modeling and Data Analysis
Denote the maximum point ymax= arg max F ( y ) . Conditions for Oly O , O
1 -.
i=l
2
1 I-, m
then equation F ( y )= 0 has the solution, 0 < y*
<
, where the function
F ( y )changes its sign from "-" to "+". ~f
F ( y m a x ) > 0 0 < ymax < 00, and
then the function F(y) has the minimum point 0 < y
< 00.
Thus, if condition (8) not valid then the equation (7) has the solution in which the likelihood function obtains minimum only. Then likelihood function n
has not maximum. At last case we proposed to take A = -00 and a = 00 . If both conditions (8) and (9) are true then we have to take into account that equation has one additional solution corresponding to minimum of likelihood fkction (4). As an initial value for numerical procedures of solving (7) we propose
where
Three Parameter Estimation of the Weibull Distribution
95
Computer modeling results by maximal likelihood approach are given in Section 3.
2.2 Analytical estimator Simple analytical estimate of the shape parameter was proposed by [Haan, 1981, Zilinskas and Zhigliavsky, 19911:
A
a=
where m
+
4 m
00, -
,
+ 0.2.
2.3. Improved analytical estimator However empirical computer simulation shows that the estimator (1 1) is biased and this bias increases when a increases. Better results are obtained when ~ ( 1 )is changed by linear estimate of the location [Balakrishnan N. and Cohen
A., 1990, Zilinskas and Zhigliavsky, 19911:
~ r n , N= ~ ( 1 -ern ) (V(m)-71
)?
(12)
Recent Advances i n Stochastic Modeling and Data Analysis
96
where
c, =
m! After simplification we obtain
1
c, = j=1
Thus more exact estimator is given by the equation:
*
a =
where 6 is taken from (1 1). To study this equation we denote
im\i
In
-
Three Parameter Estimation of the Weibull Distribution 97
1-
1
1-
1 , x=-.
a
j=l Thus, improved estimator must satisfy equation:
*
a = f(z,a*). For the solution of this equation it is convenient to use the simple iteration method: “t+l = f ( z 3 a t L
(15)
where a0 - initial value. Studying of properties of function (13) enable us to
21
make sure that equation
“0
>
1
.
lnkl
(24) has solution if
z>-
j=1 J
. Next, if
and solution of equation (14) exists, then the
98
Recent Advances in Stochastic Modeling and Data Analysis
sequence
(15) converges to the solution
of (14) with
linear rate:
3 Computer modeling We investigate the approach developed for three-parameter Weibull distribution estimation by computer modeling. Random samples from Weibull distribution (c=l, A=O) has been simulated and parameters have been estimated for every sample. Sampling averages of estimates are presented in the Table 1 and Table 2 varying the sample size N, number of order statistics m. Number of random samples M=lOO. In Figure 1 histograms of estimators are depicted.
AE
MLE
IAE
N 1000
I
2.4254142572
I
2.0301502001
I
2.6106745229
10000
I
2.4342467693
I
2.0462364888
I
2.7187719537
20000
2.3773970971
1.9687636383
2.4980881479
50000
2.3979007379
1.9538269253
2.458 1814946
100000
2.3602375824
2.0192535483
2.5933076899
Three Parameter Estimation of the Weibull Distribution
I
m=500, q=100,
I I I
I I I
1000 10000
~~
I I
5.3418326642
~~~~~~
a=5.0, c=l, A=O
I
MLE 5.0369583816
I
AE
3.3977207549 3.4717963621
I I
IAE
4.8583516027 5.0889989867
~
20000
5.2527942058
3.449861 1165
5.0855570765
50000
5.1 191912978
3.3959926626
4.8853179053
100000
5.1535 109997
3.4021267545
4.9607077656
0
99
1
2
3
4
5
6
7
8
9
10
Fig.1. Histogram of ML estimators of a (a=2.5, 100 trials) Computer modelling results show that maximal likelihood and improved analytical estimators (MLE and IAE) tolerably evaluate shape and other parameters of Weibull distribution under appropriate choice of N and m. For small m MLE sometimes does not exists. The probability of non-existence of MLE decreases when m and N increase. For reliable evaluation of large a, more order statistics and large samples would be used. The method described above could be applied for real-life problems, for instance, for optimality testing and termination of random search algorithms.
100 Recent Advances in Stochastic Modeling and Data Analysis
4 Conclusions In this paper we present maximal likelihood (ML) and analytical estimators for the estimation of the Weibull parameters, namely, translation, scale and shape parameters using order statistics of noncensored sample. Simplified method used for MLE enables us to create a simple estimation procedure of the shape parameter by solving one-dimensional equation. Improved analytical Haan estimator has been developed. Computer simulation confirmed that MLE and improved Haan estimator allows us to estimate Weibull distribution parameters with acceptable accuracy.
5. References [Balakrishnan N., Cohen A., 19901. Balakrishnan N., Cohen A. Order Statistics & Inference: Estimation Methods. Academic Press, Inc. [Bartkute V., Sakalauskas L., 20041. Bartkute V., Sakalauskas L. Order statistics for testing optimality in stochastic optimization. // Proceedings of the 7'hInternational Conference ,,Computer Data Analysis and Modeling". Minsk, p. 128-131. [Bartolucci A. A,, Singh K. P., Bartolucci A. D., Bae S., 19991. Bartolucci A. A., Singh K. P., Bartolucci A. D., Bae S. Applying medical survival data to estimate the threeparameter Weibull distribution by the method of probability-weighted moments. Mathematics and Computers in Simulation, Vo. 48, No. 4-6, p. 385 - 392. [Blischke W. R.,1974]. Blischke W. R. On nonregular estimation. 11. Estimation of the Location Parameter of the Gamma and Weibull Distributions. Communications in Statistics, Vo. 3, p. 1109-1129. [Haan L., 19811. Haan L. Estimation of the Minimum of a Function Using Order Statistics. Journal of the American Statistical Association. Vol. 76, No. 374, p. 467-469. [Hirose H., 19911. Hirose H. Percentile point estimation in the three parameter Weibull distribution by the extended maximum likelihood estimate. Computational Statistics &Data Analysis, vol 11, p. 309-33 1. [Lemon G . H., 19751. Lemon G . H. Maximum Likelihood Estimation for the Three Parameter Weibull Distribution based on censored samples. Technometrics, vol. 17, NO. 2, p. 247-254. [Murthy D. N. P, Xie M., Hang R., 20041. Murthy D. N. P, Xie M., Hang R. Weibull Models. Wiley Series in Probability and Statistics, John Wiley & Sons, Inc. New Jersey, USA. [Rockette H., Antle Ch., Klimko L. A.,1974]. Rockette H., Antle Ch., Klimko L. Maximum Likelihood Estimation with the Weibull Model. Journal of the American Statistical Association. Vol. 69, No. 345, p. 246-249. [Zanakis S. H., Kyparisis J., 19861. Zanakis S. H., Kyparisis J. A Review of Maximum Likelihood Estimation Methods for the Three Parameter Weibull Distribution. Journal of Statistical Computation and Sirnulation, Vo.25, p. 53-73. [Zilinskas A., Zhigljavsky A., 19911. Zilinskas A., Zhigljavsky A. Methods ofthe global extreme searching. Nauka, Moscow. (In Russian). [Weibull W., 19511. Weibull W. A statistical distribution function with wide applicability. Appl Mech, 18:293-7.
CHAPTER 3
Insurance
Stochastic models for claims reserving in insurance business Tamas Falukozy', Ildiko Ibolya Vitez', and Mikl6s Arat6l 1
Department of Probability Theory and Statistics Eotvos Lorind University Budapest, Hungary (e-mail: [email protected], [email protected])
Abstract. Insurance companies have to build a reserve for their future payments which is usually done by deterministic methods giving only a point estimate. In this paper two semi-stochastic methods are presented along with a more sophisticated hierarchical Bayesian model containing MCMC technique. These models allow us to determine quantiles and confidence intervals of the reserve which can be more reliable as just a point estimate. A sort of cross-validation technique is also used to test the models. Keywords: insurance, reserving, quantiles, MCMC, cross-validation.
1
Introduction
For an insurance company it is crucial to determine the future payments - generally this amount is called 'the reserve'. This is usually done by using the expected value (EV) principle which means that the reserve gives the safety appropriate to the EV. It is obvious that this is not sufficient enough, therefore insurance companies have to build a solvency capital which provides a safety puffer above the reserves for unexpected cases. In 1999 the European Union decided to reform the regulation of capital requirements for insurance companies, however, substantial work began in 2003 (Solvency I1 project). Although it seems settled that the new regulation will deal simultaneously with the reserves and the solvency capital. The main goal is to provide safety levels for the reserve and for the capital based on stochastic modelling. In this paper the socalled IBNR reserve is examined which refers to the claims that already incurred (occured) but have not been reported to the insurance company yet. There are a lot of different reserves but in non-life business the main part of the ultimate reserve comes from the IBNR and from the outstanding claims, i.e. the claims that already incurred, are also reported to the company, but are not paid yet. Up to the present the determination of the IBNR reserve was usually done by using deterministic methods based upon the so-called run-off triangle. In this triangle the claims of the past are presented and they are separated according to the time period of incurrance and to the delay of report or payment. Throughout this paper the elements of this incremental run-off triangle are denoted with C l , ] . 102
Stochastic Models for Claims Reserving in Insurance Business
103
Most of the deterministic methods uses the cumulative run-off triangle i
Ci,(see example Figure 1).
which elements are D,,, = S=l
(Maximum) Delay of reportfpayment (+1) 1
1
2
...
...
n
Fig. 1. The structure of a cumulative run-off triangle.
There are a lot of different run-off triangles, for example, the time period can be different - we examined daily, monthly, quarterly and annual periods and their influence on the reserve. Our second goal was to develop stochastic reserving methods for IBNR claims according to the main idea of the Solvency I1 project discussed above. Two naiv stochastic models are presented first and then a more sophisticated hierarchical Bayesian model is produced which contains MCMC technique. The results of the different methods are compared and the models are also tested by using some kind of cross-validation method. We have two different data-sets received from two different Hungarian insurance companies. The first one contains household insurance contracts and covers seven development years (from 01.10.1998. to 30.09.2005., with 686 71 1 contracts, 179 618 claims, 1,3 bn HUF claim amount and 10,5 mn HUF outstanding claim amount), the second one consists of motor third party liability (MTPL) contracts and covers a four-year period (from 01.01.2000. to 31.12.2003., with 500 690 contracts, 43 392 claims, 13,5 bn HUF paid claim amount and 3,5 bn HUF outstanding claim amount). These two data-sets are actually of two different kinds. The household insurance claim data-set has a short run-off which means that the claims are reported and paid within a short period of time (approximately one year), while the MTPL data-set has a substantially longer run-off. Considering that one of our aims is to analyze how the period of the run-off triangle influences the ultimate reserve, we constructed four run-off triangles to both data-sets: with daily, monthly, quarterly and yearly periods, all of them include incurred claim numbers.
104 Recent Advances in Stochastic Modeling and Data Analysis
Claim amounts were also examined, moreover, in that case we dealt with both incurred and paid claims - in that context the results can be very different, and the so-called Munich chain-ladder method, which uses both triangles and tries to reduce the gap between the results given by using them separately, was also tested. However, in this paper we discuss only the claim number results, because the claim amount results are far more complicated and discussing and analyzing them would be very spacious.
2
Deterministic and semi-stochastic methods
2.1 The Chain-Ladder technique In this section we start with reviewing one of the most widely used reserving techniques, the chain-ladder (CL) method, then we provide two naive stochastic models, which allow us to determine quantiles of the ultimate reserve. Let Di,, denote the number of claims incurred in period i and reported until period i+j-1 (i=l,...,t, j = l , ...,n-i+l) . If there are claims not reported after n years, we denote the approximated claim number incurred in first period and reported after YE periods by D,,n+. The
Di, j+l Di, j
essential assumption of the CL technique is that the lj(i) := - rates do not depend strongly on i (the period of claim incurrence), they all are approximately I,. To calculate the 1,
= D"n+ rate (the rate of very late ~
D13n claims) we use our experience from previous periods or the outstanding m
claim number. (In the latter case I , = 1+
where
is the
Q,n
outstanding claim number referring the first period.) All the other I, rates are determined by a practical weighting of the real b(i) rates - the advantage of these formulae is that we don't need to calculate the actual i'Ji) rates, we just need the X I , ]claim numbers: 1, =
~ , , J ~ J ~ ~ ~ + ~ ~ , J ~ J- ~4 , , +~1 ~+D2,J+i + ~ +-+Dn-,,,+i ~ ~ + ~ ~ ,- j , =, 1,..., l n~ - 1l . ~ ~ ~ J ~
' '
'1.J Dz,J ".+ 'n-J.1 D1.j + D2,j -+ ...+ Dn-j,j Because of its simplicity and reliability the CL method is a very popular reserving technique, moreover, it can be shown that it is correspondent with some stochastic reserving models (see for example [Verral, 20001).
Stochastic Models for Claims Reserving in Insurance Business
105
2.2 Semi-stochastic methods As mentioned before, our aim is to develop stochastic methods which we can use to provide quantiles of the ultimate reserve. First we introduce two pretty simple methods which poorly use stochastic techniques therefore we call them semi-stochastic methods.
First of all we recall the
J+1 I, (i) := values DI
which we used to
DIJ determine the Z, ratios in the CL method. Now these Z, rates are not the quotients of the sum of the D,,J values in the columnsj andj+Z but they are i.i.d. random variables. Instead of the real distribution hereafter we use the empirical distribution of these variables which is a discrete uniform distribution over the known Z,(i) values. Now, we can follow two different ways. On the one hand we can consider the expected values of these variables and by filling up the lower triangle using these values we get the expected value of the ultimate claim number, e.g., the expected value of the ultimate reserve. By the same way we can also calculate the variances of the variables and that of the ultimate reserve. Considering that we are dealing with a pretty large data-set, we assume that the ultimate reserve has a normal distribution which parameters can be estimated by the way described right now. Once we have the expected value and variance we can easily calculate quantiles of the appropriate normal distribution. This method is denoted by UN (uniform-normal). The other way is to generate a suitable large amount of outcomes of the ultimate reserve using the uniform distributions and then we can look at the empirical quantiles as the quantiles of the ultimate reserve - this method is denoted by UG (uniform-generating). Let us be more specific about the first method. Our basic assumption is that the L, variable has a discrete uniform distribution over the set
zj (i) = Di,j+l
. Therefore
~
its expected value is given
Di,j by E(L,) =
-z*, 1
n-J
n - ~D I=1
so we have to multiply the known claim
DZ,J
numbers in c o l u m j by this value to get the unknown claim numbers in c o l u m j + l . One can note that this expected value is the sum of the h(i) rates with the same weights, i.e., the average of them, so this method exactly follows the so-called ,,link ratios with simple average" method. As mentioned above we assume that the LJ variables are independent, so we
106 Recent Advances in Stochastic Modeling and Data Analysis
c[
Dn-J+l,J p E ( L k ) ]
can calculate the ultimate claim number by
2
e'g'
J=l
. As discussed
the ultimate reserve by
above this value is exactly the same that one can get using the ,,link ratios with simple average" method. But considering the 4 rates as LJ variables we are able to determine the variance of the ultimate claim number which is equal to the variance of the ultimate reserve (because one can get the n
latter as the sum of the first and the known Z Dn - j + l , jvalue). As j=1
assumed
earlier
the
LJ
1
c*, n-J
variables
are
independent,
so
D2
and using E(L:) = n - J I = I DiiJ
we get that the variance of the
ultimate reserve is given by k=J
k=J
having the expected value and variance of the ultimate reserve and assuming a normal distribution we can easily calculate any quantile of that reserve. Let us deal with the second approach. We assume that the LJ variable has a discrete uniform distribution over the set Di,j+l
l j ( i )= -
4,j
. To
get the quantiles of the ultimate
reserve we shall calculate all the possible outcomes (having the same probability), i.e., we shall determine all the possible reserves for each row (each period of incurrence). Considering that the reserve for the row i is -(Ln-i+,.Ln-r+2 Ln-l -1) and the fact that LJ can given by Di,n-i+l ....a
have (n - j ) ! different values we have (n - l)!.(n - 2). .... I! possible outcomes for the ultimate reserve. If we deal with daily periods (which means the run-off triangle for the MTPL data-set has n=1461 rows), it can be seen that it is impossible to calculate all the possible outcomes. Therefore we generate only 1000 outcomes: from the LI, Lz, ..., Ln-l
Stochastic Models for Claims Reserving in Insurance Business
variables
we
take
a
random
sample,
then
calculate
107
the
Dl,n-l+l .(L,-,+,. Ln-,+2. ... Ln-l - 1) reserve for each period i, finally by summarizing these period-reserves we get the ultimate reserve. From these thousand values any quantile of the real ultimate reserve can be estimated.
3
The MCMC method
The methods discussed so far are deterministic or semi-stochastic. Using Markov Chain Monte Car10 technique makes it possible to build up a more sophisticated stochastic reserving method. For the different lengths of period the model is the same, only the parameters and the input are different. We give the description of the model for daily data. We decided to create a one-level model, so the parameters of the prior distributions are fixed numbers chosen in such a way that the prior distributions of the parameters are best fitted to the data. For the calculations we need the runoff triangle of incremental claims, its elements denoted by Cl,,, and we also use the vector e containing the number of contracts in the several days (or months, quarters, years). In this model we assume that the data are Poisson distributed, as we are working with positive integers. The expected value of the claims incurred on day i, reportedj-2 day later is the product of e,, the number of (valid) contracts on day i, X,, the claim intensity of day i per one policy, and 5, the proportion of claims reported j - l days late: E(C,,, ) = elX,Y, , (i, j = 1,...,u ) . Assuming that the claim intensity of day i, X, and the proportion of claims reportedj-l days late, 5 are random variables, we have to determine their prior distributions. XI,...X n are considered to be prior independent and identically Gamma distributed random variables. We also fitted the model with other prior distributions such as normal but this choice seemed to be the best working. Additionally, when examining claim numbers it is widespread to use Poisson distribution for the number of claims and Gamma for the intensity as this way the unconditional distribution of claims incurred on a fixed day is negative binomial. ,.., Y, are prior Beta distributed random variables on the appropriate intervals, determined below. We have the n
constraint X Y , = 1 as we suppose (presume) that all the claims are k=l
reported at most n days late. The parameters of the prior distributions are chosen as follows. Alpha and lambda parameters of the Gamma
108 Recent Advances in Stochastic Modeling and Data Analysis
distribution are calculated in such a way that the expected value is the mean of the X i-s' starting values, and the variance is suitably high to let the data dictate. For the starting values of X , we used the naive estimate:
vectors C,!,,= C, /e,. The prior distribution of ,
Y, (i = 1,..A- 1) is Beta
with parameters a, b where the variance is fairly high and the expected value is greater than the half length of the interval. Thus, we have the following prior distributions:
where notation [x I y ] denotes the conditional density function
f,.,, (xly)
in continuous cases, and the P(X = x I Y = y ) conditional probability in discrete cases, respectively. Providing that the elements C,,, have Poisson distribution with the appropriate parameter, the joint posterior distribution is:
We obtain the posterior distribution of
X,:
Stochastic Models for Claims Reserving in Insurance Business
The posterior density function of
109
is a bit more complicated:
Clearly, the latter is not the density function of a well-known distribution so we have to use Markov Chain Monte Carlo simulation to get the expected values and also the appropriate quantiles. We determine certain quantiles of the reserve in the following way: in each MCMC iteration we generate a subsequent sample of X, and Yj, and also ofC,,, as random sample of Poisson( e,X,Y, ). So in each iteration we have a whole square of claims incurred in the first n days and reported at the most n-I days late fiom which it is easy to calculate the reserve as the sum of the values in the right down triangle. Therefore we performed 1 000 000 iteration of the MCMC method, leaving the first 100 000 out as the burn-in period, using the rest for determining the point-estimate and the quantiles. In each iteration we obtained a subsequent sample of the parameters and using them we generated a Poisson-distributed value with the appropriate expected value for the reserve. This way we obtained the quantiles deriving from these 900 000 values. 4
The results
In the previous sections we introduced the models used to estimate the reserve of the two data-sets, here we present the results for the monthly and the quarterly data - daily periods result a high variance when determining the quantiles and using annual data we have just four or seien rows in the run-off triangle which is insufficient for a correct MCMC algorithm.
Fig. 2. Point estimates of the reserve for the household data-set.
110 Recent Advances in Stochastic Modeling and Data Analysis
Fig. 3. Point estimates of the reserve for the MTPL data-set.
Figures 2 and 3 show the point estimates for the reserve given by the different methods and by using different time periods for the household and MTPL data-set, respectively. Of course we received different values working with the different periods but we also got highly different results using the various methods on the same run-off triangles which is not so obvious. It can be seen that the estimates of each algorithm grow with the length of the time period. As for the different methods it seems that the measures produced by MCMC method are consequently near the chainladder estimates while the two semi-stochastic methods are similar to each other, always giving higher results than the other two - as far as the household data-set is concerned. But considering the MTPL data it is the MCMC method that gives different values which are lower than the estimates of the CL technique while the two simple stochastic methods result very similar values. Therefore it looks like that choosing the best method can be done only according to the data.
Fig. 4. Quantiles of the reserve for the household data-set.
Stochastic Models f o r Claims Reserving i n Insurance Business
111
'i
Fig. 5. Quantiles of the reserve for the MTPL data-set.
For the quantiles we have three methods as from the chain-ladder technique we get only a point estimate. In Figures 4 and 5 the 50, 75, 90, 95% quantiles for the reserve can be seen for the three different methods using different time periods. As previously stated the MCMC results tend to be lower than the others and this is true for all quantiles and for both data-sets. This method differs from the other two in respect of the variation which is basically lower than that of the semi-stochastic methods. As mentioned before, the daily data give always the highest variance, but for the other periods the variance is neither decreasing nor increasing with the length of the time period. 5
Testing
The purpose of this paper is to calculate the reserve for an insurance company's claim numbers. In the previous chapters we introduced various methods of estimation, and of course we are interested in comparing these methods, let alone the fact that our estimates are rather different. A way of testing the goodness is calculating these estimations for just the first part of our data, and the real value of the reserve using the continuation of the data. By comparing the estimates to this last value it turns out how good the different methods are for this sequence of data (a sort of crossvalidation). It is unnecessary to explain how important is to know more about the reliability of these algorithms. The result of our calculations show that even in the simple case of household claims some of the basic procedures (like chain-ladder) fails. (As the MTPL data-set covers only four years, this testing method can barely applied to it.) For the test calculation we used the claims incurred and reported in the first 3 years. Here we present the results received by using monthly periods. The real
112
Recent Advances in Stochastic Modeling and Data Analysis
value for this part of the data is 1788 while the CL method gives only 990 which is a really bad underestimation.
Fig. 6. Quantiles of the reserve for the test data-set compared to the real value.
Figure 6 shows the quantiles of the reserve for this first three years computed by the three stochastic methods compared to the real value. Even if the point estimate is low, the algorithm can be acceptable if the real value of the reserve lies in the [lo%, 90%] confidence interval of our estimations. As discussed above the MCMC method gives the lowest results and has the lowest variance so it is not surprising that all the quantiles given by this model lie under the real value. But the 90% and 95% quantiles of the two semi-stochastic methods exceed the real value (in the case of the UN model also the 75% quantile is higher) so by using these methods it is possible to get reliable estimates for the reserve.
6
Conclusion
As a conclusion we can say that longer time periods result higher reserves, while using more detailed data (shorter periods) lowers the result but also raises the variance and therefore the uncertainty. Concerning the other goal, i.e., to develop stochastic models for claims reserving (it is obvious that the point estimates given by the deterministic methods are not reliable enough), one can see that also the stochastic models can provide very different results. The main question is how to choose between these models. We suggest that one should apply and also test all the methods (the deterministic models too), and then the most appropriate can be taken according to the type of the data set (i.e. shorter or longer run-off) and to the results of testing. In the future we want to develop our methods by using bootstrap technique and compare them with other stochastic models.
Stochastic Models for Claims Reserving in Insurance Business
113
References [de Alba, 2002lE. de Alba. Bayesian estimation of outstanding claim reserves. North American Actuarial Journal, volume 6, number 4, pages 1-20. [de Alba, 2004lE. de Aha. Bayesian claims reserving. Encyclopedia of Actuarial Science, John Wiley and Sons, Ltd., Sussex, UK. [England, 2002lP.D. England. Addendum to “Analytic and bootstrap estimates of prediction errors in claims reserving”. Insurance: Mathematics and Economics, volume 3 1, pages 46 1-466. [England and Verrall, 1999lP.D. England and R.J. Verrall. Analytic and bootstrap estimates of prediction errors in claims reserving. Insurance: Mathematics and Economics, volume 25, pages 281-293. [England and Verrall, 2002lP.D. England and R.J. Verrall. Stochastic Claims Reserving in General Insurance. Institute ofActuaries and Faculty ofActuaries, pages 1-76. [Mack, 1994lT. Mack. Which Stochastic Model is Underlying the Chain Ladder Method?. Insurance: Mathematics and Economics, volume 15, pages 133-138. [Mack and Venter, 2000lT. Mack and G. Venter. A comparison of stochastic models that reproduce chain-ladder reserve estimates. Insurance: Mathematics and Economics, volume 26, pages 101-107. [Renshaw and Verrall, 1998lA.E. Renshaw and R.J. Verrall. A stochastic model underlying the chain-ladder technique. British Actuarial Journal, volume 4, number 4, pages 903-923(21). [Verrall, 2000lR.J. Verrall. An investigation into stochastic claims reserving models and the chain-ladder technique. Insurance: Mathematics and Economics, volume 26, pages 91-99. [Gyarmati-Szabo and Markus, 200715. Gyarmati-Szab6 and L. Mhrkus. A Hierarchical Bayesian model to predict belatedly reported claims in insurances. Manuscript, 2007.
Stochastic Risk Capital Model for Insurance Company Gaida Pettere Banking Institution of Higher Education K. Valdemara street 1b, Riga LV 18 19, Latvia e-mail: gaidaB1atnet.h Abstract. The paper is about stochastic modelling risk capital requirements to cover equity risk for an insurance company by using copulas. We have tried to find the best portfolio structure with copula model and simulation using risk measure VAR. The result is compared with equity shock model given in Solvency I1 documents.
Keywords:equity risk, portfolio optimisation, copula model, Value-at-Risk, solvency
1 INTRODUCTION All insurance companies in Europe are starting to follow Solvency I1 requirements. One of the main approaches of Solvency II is solvency capital requirements for insurance companies. The Value-at-Risk has become very important risk measure because it expresses the amount of capital to cover certain amount of losses with some probability. Usually probability of insolvency is chosen arbitrary and remains around 0.5% if we are talking about loss distributions. VAR as risk measure for all insurance risks has become main keystone in evaluating capital requirements under Solvency I1 approach. In the same time it is advised to evaluate equity risk as part of market risk to assume 32% decrease for global equities and 45% decrease for other equities without taking into account diversification of portfolio (see CEIOPS’ materials). Such simplified assumption leads to very high solvency capital requirement for equity risk. Therefore our aim is to show the difference between the mentioned simple assumption and stochastic approach in solvency capital calculations using as risk measure the same VAR measure as for insurance risks. There have been many attempts to evaluate risk capital in recent years (see Panjer (2003), Bagarry (2006), Fabien (2003)) but all of them deal with multivariate Gaussian or elliptical distributions for modelling loss distributions. VAR as a risk measure in asset pricing and portfolio optimization is used very widely in last ten years despite that VAR and expected shortfall (ES)as possible risk measures were already introduced by Markowitz (1959). In last 15 years many publications have appeared about using copulas in different fields of our life. The advantage of copulas is possibility 114
Stochastic Risk Capital Model for Insurance Company
115
to estimate VAR of portfolio in the case of non-Gaussian and non-elliptical distributions of portfolio. Breymann, Dias and Embrechts (2003) have shown how to use copulas when working with high frequency data in finance in the two-dimensional case. Later Embrechts, Hoing and Juri (2003) have found estimators of upper and lower bounds of VARfor portfolio consisting from the equal parts of n equities and have given two-dimensional examples. More widely copula methods in finance are used by Cherubini, Luciano, Vecchiato (2004). They have shown how to simulate multivariate n-dimensional copulas and how to use copula technique in VAR estimation for equity portfolio. But only two-dimensional cases are considered as examples. Embrechts and Hoing (2006) have found how to estimate V&? for ndimensional portfolio which consists of assets in equal amounts using copulas. We have tried to work out model how to find high level VAR of portfolio if it does not consist from the same amount of equities. Necessity for that is discussed by Embrechts and Hoing (2006). At the same time we have tried to find the best proportion of equities in portfolio to have smallest necessary solvency capital in the case when no marginals no portfolio distribution are elliptical.
2 METHODOLOGY We have investigated portfolio consisting of three equities with correlation around 0,5 and less. We put together the portfolio from equities of American Express Co (Axp),Walt Disney Co (DIS) and Hewlett Packard Co (HPQ) with correlation coefficients = 0.32, = 0.53 andp2,3= 0.37, where index 1 is used for rate of returns of American Express Co, index 2 for Walt Disney Co and index 3 for Hewlett Packard Co. We calculated rate of returns by formula:
Ri= r::+Wdqys - 1:
r:
where is the price of the ith security. One quarter (90 days) is used as a time lag because our liabilities are calculated on quarter basis. Rates of return were transformed later (transformed rates of return) to have losses OR right hand side:
xi=1-E;5.
Portfolio is considered consisting from mentioned securities in different proportions:
116 Recent Advances in Stochastic Modeling and Data Analysis
cai .xi 3
xp,t+T =
i=l
where
cai 3
with the condition that
= 1,
i=l
ai is the proportion of the ith security in portfolio
We have used in our investigation two-stage method: at first we estimated the parameters of marginals by using Kolmogorov D (absolute value of difference of empirical and theoretical distribution hnctions), secondly we estimated chosen copula parameters applying numerical method for minimizing sum of squared distance between sample and simulated data. Several classes of multivariate Archimedean copulas: Clayton, Frank, survival, two-parameter copula (Nelsen, 1999) and Gaussian were investigated. Kendall’s Tau was used as the dependence measure. After appropriate copula was found we 50 times simulated 50000 data, constructed portfolio using formula (3) and found its empirical VAR(99.5%). After that we started to investigate the behaviour of VAR to change proportion of securities in portfolio. All calculations have been made using Mathcad and Microsoft Excel software.
3 DISCUSSION AND RESULTS The data under consideration consists of 1165 rates of returns from 01.06.2001 until 2.06.2006 with time lag 90 days for each security calculated by formula (3) and transformed by formula (4). Basic characteristics of all three random variables are presented in the following Table 1 and correlations in Table 2. Table 1. Characteristics of examined random variables
Kurtosis Skewness Sample size ‘Largest(1) Smallest(1)
0.39515 -0.04863 1165 1.37041 0.55841
-0.14765 0.55240 1165 1.43354 0.60120
-0.19225 -0.10848 1165 1.40796 0.29948
hStochastic Risk Capital Model for Insurance Company 117
Kendall’s Tau
Linear correlation coefficient
AXP
1
DIS
HPQ
0.321 1
0.528 0.371 1
1
0.544 1
0.310 0.357 1
First the marginal univariate distributions are examined using families of lognormal and Wald distributions. Random variable X has Wald distribution with parameters p and 2, if the density is of the form (Balakrishnan, Nevzorov, 2003):
We have applied Kolmogorov goodness of fit test to find the best approximations for all three random variables. Used parameters, and the test statistics for comparison with the 5% critical value of Kolmogorov test (0.039845) are shown in next table (Table 3).
Both distributions give good approximations as it is possible to see to compare calculated values of D with the critical value of the Kolmogorov test statistic. Graph of the empirical distribution and both theoretical densities for transformed rate of returns of American Express Co are shown in Figure 1. We have used standard algorithm for Archemedian copula generating described in Frees, Valdez (1998) and later for Clayton, Gumbel and Frank copulas a generalized version by Cherubini, Luciano, Vecchiato (2004): 1. Generate U,, ,..., independent uniform (0,1) random
u, up
numbers.
118 Recent Advances in Stochastic Modeling and Data Analysis
2.
Set XI = 4-'(UI) and co = 0 .
3.
For k = 2, . . . , p , recursively calculate X k as the solution of
where p is the generator of an Archimedean copula, q
-l(k-I)
is the k - 1
order derivative of the inverse generator, Fk- marginal distribution hnction and 'k
0.1
1
= d6(x1>1+ d 4 (
x 2
>I+..*+p[& ( x k
1
0-
Figure 1. Approximation of transformed rate of returns of American Express Co by lognormal and Wald distribution. We have worked out ourselves the mentioned simulation algorithm for Survival copula and for two parameter Copula 4.24 from Nelsen (1999), p. 124. Algorithm for Survival copula is the following: 0 Simulate 3 independent random variables u l , u 2 , U3 from
W O , 1) 0 0
Set XI = e-I(u1) Calculate C1= ln(1- 6 In u l ) Find 1-(1-61n(F2 (x2)))eC'
1-eC1 ~
x2 t root[(1 - 6 In( F2(x2)))e
6
-242-e
,x2]
Stochastic Risk Capital Model f o r Insurance Company 119
Calculate C2 = ln[(l - S In ul)(l - S In u2)] I-(I-JI~(F,(x~)))~'~ 0
Find x3 t root[(l- 8 h(F3(x3)))e
6
The algorithm for Copula 4.24 is: 0 Simulate 3 independent random variables ul, u2, U3 from
W O , 1)
'
0
set xl = e - l ( u 1 ) .
0
Calculate C, = (uI-" -1)'.
0
Calculate C, = (ul-"
- 1)' + (u2-" - I)'.
x3 t root[((c2+ ((F,(x3))-"- 1)"F
1
-1-a
+ 1)"
1
x(c2 +((F,(x3))-"-l)p>-2+p
We have used also the standard algorithm to simulate Gaussian copula (see Cherubini, Luciano, Vecchiato (2004)). We investigated all mentioned copulas (Clayton copula, Frank copula, Survival copula, Copula 4.24 (Nelsen (1999) example 4.24)) with lognormal and Wald distributed marginals. Copula parameter and the copula itself for hrther applying was found by using the fitting measure
cc H
sEE=
H
k=l J = 1
H
(Oi, j , k -,i' i=l
H3
j,k
1'
120 Recent Advances in Stochastic Modeling and Data Analysis
where maximum lengths of each axis are divided into H segments and ff 3 rectangles created, Oi,j,kis observed frequency in each rectangle and Si,j,k is the simulated frequency in each rectangle under the condition that size of the simulated data coincides with data size. Average values of the fitting measure and its standard deviation with the best copula parameter are shown in Table 4. Marginals
Clayton copula
Frank copula
S = 1.4 Lognormal, lognormal, lognormal Wald, Wald' Wald
Survival copula
6 = 7.5
p = 12.256 0 = 0.782 p = 7.486 0 = 0.578
p = 8.470 0 = 0.489 p = 8.903 0 = 0.563
Copula 4.24
S=-O.1
a=5, ~=1.4
p =20.156 0 = 1.109 p = 19.023 0 = 0.989
p = 13.024 0 = 0.566 p = 14.356 0 = 0.675
Gaussian copula
p = 16.34 0 = 0.967
As it is possible to see from Table 3 the best approximation is reached with Clayton copula with Wald marginals. Finally we have simulated 50 000 data 50 times with that copula, have calculated portfolio with different proportions of securities in it and have found average of empirical VAR(99.5%) of portfolio. Results with several proportions of securities are shown in the Table 5. As it is possible to see from mentioned table, it is possible to find optimal structure of portfolio that gives better 99.5% VAR to compare with 32% or 45% decrease. Table 5. Portfolio VAR and corresponding rate of return in comparison with
0,4 0.2
I I
0,2 0.2
I I
0,4 0.4
I I
1.221 1.395
I
I
-0.221 -0.395
I
I
0 0.6
I
I
0.4
0.4
I
I
0.6 0
I
I
1.106
1.201
I
I
-0.106
-0.201
4 CONCLUSIONS Solvency I1 papers claim that no diversification can be taken into account but in the same papers is advised to add all risks with correlation matrix. Like it is possible to see from Table 4 diversification plays large role in solvency capital determination. The negative point is that it is difficult to
Stochastic Risk Capital Model for Insurance Company
121
find appropriate copula because most widely used copulas have only one parameter and it is difficult to fit the distribution to data. Even two-parameter Copula 4.24 did not give better approximation..
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Bagany M. (2006) Economic capital: a plea for the Student copula. 28"' International Congress of Actuaries, http://papers.ica2O06.com/30 14.html. Balakrishan, N., Nevzorov, V. (2003) B. A Primer on Statistical Dism'butions.Wiley, New York. Breymann W., Dias A,, Embrechts P. (2003) Dependence Stsuctures for Multivariate High-Frequency Data I Finance, Quantitative Finance 3(1) 116, m.math.ethz.ch/-baltes/ftp/papers.html. CEIOPS'advice to European Commission in the light of Consultation Paper CP15-20, m.gcactuaries.org, Solvency I1 page. ceiops_qis2_final,www.pcactuaries.org, Solvency I1 page. Cherubini, U., Luciano, E. (2001) Value at risk trade-off and capital allocation with copulas. Econ. Notes, 2001, v01.30,235-256. Cherubini U., Luciano E., Vecchiato W. (2004) Copula Methods in Finance, Wiley, New York. Embrechts P., Hoing A., Juri A.(2003) Using Copulae to bound the valueat-Risk for functions of dependent risks, Finance dt Stochastics 7(2) 145167, www .math.ethz.ch/-baltes/ftp/papers .html Embrechts P., Hoing A. (2006) Extreme VAR scenarios in higher dimensions, www.math.ethz.ch/-baltes/ftp/DaDers.html. Fabien F.(2003) Copula: A New Vision forEconomic Capital and Application to a Four Line of Business Company, www.actuaries.org/ASTIN/Co~lo~uia/Berliil/F Frees, E. W., Valdez, E. A. (1998) Understanding relationships using copulas. North American Actuarial Journal, vol. 2, 1-25. Hurlimann W. (2002) An alternative approach to portfolio selection, in Proceedings of the 12"' international AFIR Colloquium, Cancun, Mexico. Markowitz, H. (1959) Porvolio Selection. Wiley, New York. Micocci M., Masala G. Basktesting value-at-risk estimation with non Gaussian marginals. Nelsen, R B.(1999) An Introduction to Copulas. Springer-Verlag, New York. Panjer H H. (2003) Risk, Solvency and Allocations of Capital in Financial Institutions. Financial Engineering News, 2003, vol. 30, MarcldApril, www.fenews.codfen30.
Measuring demographic uncertainty via actuarial indexes Mariarosaria Coppola' , Eniilia Di Lorenzo', Albiiia Orlando3, and Marileiia Sibillo4
*
University of Naples Federico I1 via Rodin6 22 80134 Napoli, Italy (e-mail: m. coppola0unina. it) University of Naples Federiro I1 via Cintia, Complesso Monte S. Angelo 80126 Naples, Italy (e-mail: diloremiounina. it) CNR, Istituto per le Applicazioni del Calcolo Mauro Picone via P. Castellino 111 801213 Naples, Italy (e-mail: a. orlando0iac. cnr . it) University of Salerno via Ponte Don Melillo 84084 Fisciano (SA), Italy (e-mail: msibilloounisa. it)
Abstract. Aim of the paper is the analysis of the behaviour of risk filters connected to the demographic risk drivers for a portfolio of life annuities. The model, easily suitable to the rase of pension annuities, involves the evolution in time of the mortality rates, taking into account the randomness of the financial scenario. Within this context, the uncertainty in the choice of the deiriograpllic scenario is measured and the analysis is also supported by the VaR sensitivity to this risk source. Keywords: Longevit,y risk, Value at Risk, Risk index.
1 Solvency assessment principles: the importance of the longevity risk 1.1
Basic guidelines on solvency
In a solvency assessnient framework the keystone is represented by the capability of assets to cover obligations. Of course such valuations can be pursued on a run-off basis, i.e. considering only written business, or as an on-going concern, i.e. considering also future new business. As explained by [Cocozzaet al., 2006a], several a.pproaches have been considered in literature and in practice, for instance consistently with a surplus analysis, according t o an actuarial perspective, or with an income valuation. according to an accountant perspective, or with a profit analysis, according t o an ecoiioinic point of
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view. In any case, whatsever valuation perspect.ive one chooses (that can be aimed, for instance, t o the financial policy of the company or whatever company's policy), it is quite iinportant what follows (cf. [Cocozzaet d., ZOOCia] for a deeper understanding): 0 0
0
identifying the risk drivers affecting solvency, iiieasuring these risk sources, defining the capital requirements apt to cover liabilities.
Categorizing any possible risk must be opportunely addressed t o a specific purpose, ta.king into account several points of view: the shareholder's, the policyholder's, the supervisory a,uthority's ones. Focusing on a run-off basis; according to a. supervisory authority perspective, two risk components basically affect a life insurance system, that is the demographic and the financial risk drivers. Demographic risks are due to the differences between the anticipated mortality rates and the actual death frequencies; analogously financial risks are due to the differences between the actual return on assets (obtained investing premiums or contributions) and the interest rates adopted in the technical bases. Capital requirements can he based on fixed ratio systems or risk-based approaches. As well known, the first procedure, structured on a fixed percentage of a quantity representative of the risk exposure, which determines the solvency margin, does not capture the company's pecularities in terms of risk profile. The risk-ba.sed a.pproa.ch is cha.ra,cterized by connecting ca.pita.1 reqiiirement's to the specific risk profile of t,he insiirance business, that) is (cf. [Cocozzaet al., 2006bl) asset risk, insurance risk, interest ra.te risk, business risk (see RBC requirements developed by NAIC). On the other ha.nd, according to the Ra.sle indica.tions., iiiterna.1 models are realized connecting the siirpliis level t o the company's results and linking capital requirements to ruin probabilities. This approach is framed into a stochastic model involving an overall valuation perspectivc, apt, t,o assess solvency in connection t o the company's activities. In general solvency can be assessed on the basis of the ineqiialit,y:
q . 4 , - Lt 2 0 ) = E ,
E > O
where At and Lt. are; respectively, assets and liabilities at time t . Within a risk mapping procedure, in the following we analyze the impact on the liabilities of the systema.tic deniogra.phic risk, ta,kiiig into a.ccount its intera.ction with the financial risk soi.irces. In this order of ideas the int,ernal risk profile is deeply investigated by means of suitable risk indexes, apt to describe the impact of the selected risk source, as well as additional costs can be estimated with a fixed reliability degree.
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1.2
An overview on longevity risk: the dynamic aspect of mortality and the projected survival models
The traditional survival models describe the age behaviour of mortality in a wide period, from age 0 t,o the maximum att,ainable age, attributing to the force of mortality, thereby to the probabilities of death, values deriving from period iiiortality observations. Such models assume the mortality trend static in time and manifest itself t o be inadequate in capturing the betterment in the mortality trend, known as the longevity phenomenon, particularly observed beginning from the past century in all industrialized countries (cf. [Pita.cco 20041). The curves of dea,ths reveal to be strongly clia.ra.cterizedby the increasing concent.rat#ionin the inode (recta,ngi~larization)and at the same time by the mode random approaching toward very old ages (expansion), effecting decreasing mortality profiles. In substance, the evolut,ion in time of the number of deaths presents, besides the xcidental deviations from the expected values, systematic deviations too. In actuarial calculations, period-based assuinptioiis for siirvival probabilities mean to represent the future as the past (even if recent past), without taking into account that future death probabilities will he most likely less than the corresponding period-ba,sed ones. In the fair value framework, the question of the correct choice of tools modelling the mortality pattern becomes noda.1 observing that no indications about mortality systematic deviations from the expected values come out from the market, so that values are obtained as marked to model values. Otherwise the longevity bond market is not perfectly developed yet for the incompleteness of the a.ges a.t issue represented in the contra.cts a.nd for the aspect of the incomplete liquidity. As suggested in [Ballottaet al., 2006], in the current valuation approach the deinogmphic phenomenon can be represented a.s the expecta.tion of its best estimate, considering the market neutral with respect to the two aspects, systematic and unsystematic, of the demographic risk. In this basic assumption, the. projec morta.lity tables., founded on models capt'iiring t,he dynamic aspects of the human life, in particular with respect to longevity, coiistitute the tool for fronting the problem. The consequence in the risk mapping is tha.t the demographic risk component results from the combined action of the mortality risk, deriving from the accideiital deviations of the number of deaths from the expected value, and the risk arising froin the use of a. ta.ble not a.dequa.te in representing the future mortality, the demographic m,odel risk. The demographic model risk is studied in the death benefit context in [Olivieri , 20011, in which term a,ssura,nce portfolios are considered paying att,eiition to new deceases. In [Coppola et al., 2006b], the impact, of mortality ta,bles with different degree of projection is analysed with respect to the fair valuation of the mathematical provision in t,he part,icular case of a term assurance with decreasing capital. In that paper the influence of the de-
Measuring Demographic Uncertainty via Actuarial Indexes
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mographic model risk on the periodic premium duration is practically shown and measured. On the other side, due to their long dura.tion a,nd ma.nifold payments, the actuarial appraisal involving life annuities is indeed a business section heavily affected by the demographic model risk. The choice of a mortality table with an iiiifair degree. of projection can cause remarkable iinderestiination of future costs, including insurers’ or pension fund liabilities or, otherwise, too high amounts constrained for reserves. In light of these considerations, in the life benefit contract framework, the longevity component holds a primary role in the insura.nce business risk ina.na.geiiient a.nd in the solvency a.ssessment. In papers by [Olivieri , 20011 and [Coppolaet al., 20021, the longevity risk is shown to be independent on the portfolio size and the model risk is introduced and measured outlying different inort,ality scenarios. Di Lorenzo and Sibillo (cf. [Di Lorenzoet al., 20021 draw the behaviour of the this risk for the mathematical provisions of a life annuity portfolio as fiiiict,ion of age at issue and portfolio size, picking out a critical age in correspondence of which the projection risk changes its trend. In the same paper some considerations arising from the actuarial functions’ pattern are developed. The relative importance of the two demogra.phic risk coiiipoiients, na.mely the insura.nce risk and the model risk, are coinpared in [Coppolaet d., 2002], together with the contribute of the stochastic interest rate to the global portfolio risk.
2
The mathematical provision fair value in living benefit products
The risk analysis we propose is based on stochastic assumptions both for the financial coinponeiit, due t o the random movements of the interest rates, and for the demographic one. The stochastic framework we will work in is outlined in its evolution in time by the information flow containing the financial and the demographic information formalized in the two probability F’, P’), (0, F”, P”), where F’ and F” a.re the a-a.1gebra.sreferred spa.ces (0, t o the financial and survival information flow respectively; we suppose the independence of the randomness in interest rates on the randomness in mortality rates, as commonly assumed. The two spaces generate the probability space (Q, F, P ) , represented by the filtration { F k } C F , with Fk = FL U 8‘: and {FL} C F’, { F ’ k } C F . The financial market in which the valuation is framed is assumed to be frictionless, with continuous trading, no restrictions on borrowing or short sales, the zero bonds and the stocks being iiifinitely divisible. We consider the case of payments due in case of life at the end of each period to persons belonging t o a.n initia.1 group of c coeval individuals aged 2. If w ( t , j ) is the market value at t,iine t of one monetary unit! due at time j and X j is the stochastic cash flow at time j , we can write the fair value at, t,iine t of the stochastic stream of loss from t to the coiitract,ual ending time:
126 Recent Advances in Stochastic Modeling and Data Analysis
In formula (1) Nj indicates the number of claims at time j in a portfolio of hoinogeneous policies, coinciding with the iiuinber of survivors at that time, in the case of life annuities. In (1) the generic element of the trading strategy replicating the portfolio flow at time j is constituted by N J X , units of unitary zero coupon bond issued in t and maturing in J with the following price (see [Coppola. et al., 2006bl):
The operator E in formula (1) denotes t,he expected value under a risk neutral probability measure hypotheses, deriving by the completeness of the market. This assumption, acceptable in relation to the financial aspect of the operation, is less realistic for its demographic component. As suggested in [Ba.llotta.et al., 20061, the expected va.lue under a. risk neutra.1 proba.bility measure can be calculated using the most opportune probability measure for the demographic component, taking into account both the systematic and the unsystematic mortality risks. In the case of an immediate life annuity portfolio, following [Coppola et al., 2006a], having iiidicated by t p , the probability that the individual aged z survives at the age 2 t , we write:
+
3
Risk indexes in living benefit contract fair valuation
Measuring the impact of the demographic model risk on a fiiiaiicial undertaking involving life benefit payments means quantifying its variability due to t,he raidomness in the. choice of the mortality table, the effects of the other two risk components (stochastic interest rates and random deviations of mortality) having been averaged out (cf . [Coppolaet ul., 20021). Referring to the case considered in section 2, we measure the impact of the demographic model risk ( D M R M ) on the value V, calculated in formula (3), proposing the. following expression (see [Coppola et al., 2006aI)
DMRM
= Va~[IE[v,lKtl] = c ’ V a r [ E [ x Xjjp,jp,.+ttJ(t,
j)lKt]].
(4)
j>t
having estimated by the conditioning on h’t the randomness in the choice of the survival function used to determine the survival probabilities.
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Measuring Demographic Uncertainty via Actuarial Indexes
Numerical applications
4
First of all we estimate the risk index DMRM in forinula (4),in the case of a portfolio of c = 1000 immediate unitary life annuities, a,ccording to a demographic scenario represented by the mortality rates involved by 0
0 0
the Italian inale life tables S I M 2002, the Weibul model with parameters cy = 85.2 and y = 9.15, the Lee-Ca,rter model a.pplied to the 1ta.lia.nda.ta. (referred to the Ita.lian popillation from 1947 to 1999 and from age 0 to 109, where 109 is the limiting age).
with reliability degrees, respectively, 0.2, 0.3, 0.5. The first table is the less projected one a.nd the third ta.ble the most projected. The fiiiancia.1sceimrio is represented by a CIK process for the interest rates, adjusted on daily average yields on fixed interest securities over the period January 1999- July 2006. t=5
t-10
t=l5
t=20
t=25
t=40
t=50
z = 40 984,8022 1,078,248 1,542,4113 1,984,756 2,480,825 3,156,176 1,278,737 0 x = 65 1,198,026 1,429,125 1,285,527 1,200,000 410,178.8 0.01886
Table 1. Projection Risk -(Model risk) estimation, reliability degrees: 0.2, 0.3, 0.5
Consistently to the results obtained in [Di Loreiizod al., 20021, t.he values in Table 1 show that the projection risk, for every fixed age at issue, generally increases with the time of the reserve valuat.ion, until the age reached at that time assumes a certain (high) value (depending on the pecularities of the selected tables). This valuation time is higher the younger the insured is. The results obtained in Table 1 reflect the reliability degree assigned t o each survival model, as the results in Table 2 confirm, where the probabilities given t o each survival table are, respectively, 0.2, 0.6, 0.2. t=5
t-10
t=15
t=20
t-25
t=40
t=50
z = 40 616,076.6 745,656.3 952,733,3 1,217,554 1,516,759 1,951,206 800,480.4
z = 65 733,151.5 874,301.6 790,432.4
741,963 253,750.6
0.278133
0
Table 2. Projection Risk -(Model risk) estimation, reliability degrees: 0.2, O.G, 0.2
It is interesting t o observe tha.t for the insured a.ged LL: = 40, the risk index, capturing the projection uncert,aint,y, increases until a valuation t,iine greater than the time in which 5 = 65 reaches his hit, since the results incorporate the behaviour of t,he Lee-Carter survival probabilities, where t,he survival dynamic in time reflects the starting age. In confirmation of this, we observe in
128 Recent Advances in Stochastic Modeling and D ata Analysis
Table 3 the behavioiir of t,he va,lues, corresponding t,o the ages z = 40, z = 60 and z = 65, using the Weibull model with parameters ( a = 83.5;g = 8), ( a = 85.32; 8 = 9.15), ( a = 87.0; ,/? = 10.45) and reliability degree, respectively, 0.2, 0.6, 0.2. In fact in this case the three reserves corresponding t o the ages z = 40, z = 60 and z = 65 begin t o decrease when the corresponding insured is between the ages 65 and 70. The behaviour for older ages is due t o the intera.ctions between z a,nd t rela.ted to the projected probability t p , and the residual duration of the contract (cf. also [Di Lorenzoet al., 20021).
t=5 t-10 t=15 t=20 t=25 t=30 t=40 t=50 z = 40 303,495.2 478,901.2 493,899.7 547,731.1 552,953, 1 480376.3 159,494.5 4,576.932 z = 60 421,733.1 406,077.3 279,729 138,056.7 37,788.14 18.73209814 12.1783 0 z = 65 188.557.7 139,OG7.2 118, GG7.9 32.894.78 3.1G0.081 0.223 0.048717 0
Table 3. Projection Risk -(Model risk) estimation, Weibul model
The following tables reporte the quantile reserve values obtained by means of a siiriulatioii procedure developed in [Orlandoet al.. 20061. Here, in order t o have a better approximation of each simulated path for the interest rate, we consider a weekly sample interval. t=5 t-10 t=15 t=20 t=25 t=40 t=50 SIM2002 21,278.48 18,516.65 15,891.98 13,495.11 10,737.45 4,649.763 1,759.868 Weibul 22.218.74 19.370.08 16.637.88 14.086.61 11.225.09 5.021.374 1.818.434
Table 4. Quantile Reserve. Confidence level=95%, x=40 c= 1000
t=40 t=50 t=5 t=lO t=15 t=20 t=25 SIM2002 21,285.61 18,522.31 15,896.32 13,498.28 10,739.62 4,650.199 1,759.966 Weibul 22,226.47 19,376.21 16,642.60 14,090.07 11,227.46 5,021.893 1,818.523 LeeCarter 23.816.98 21,210.78 18,839.69 16,726.03 14,199.61 10,524.92 4,698,106 Table 5. Quantile Reserve. Confidence level=99%, x=40 c=1000
We see that, if the projection degree increases, the quantile reserve for both the considered confidence levels increases. This trend is stronger in the case of the Lee-Carter table and when the t,iine of valuation is high. Finally, all the numerical results shows that, in a solvency perspective, it is
Measuring Demographic Uncertainty via Actuarial Indexes
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opport,iiiie to take into account context,iially different kinds of risk indexes. in particular referred t o the uncertainty in t h e choice of survival tables a n d to t h e quantification of t h e fiuancial position for different tables. In fact, t h e risk index i n (4)provides a measure of t h e overall effect of t h e uncertainty i n the choice of t h e deiriographic scenario. whilst t h e evolution i n time of t h e quantile reserves is a concrete indicator, in terms of fiiiancial results, of the a.bove choice.
References [Ballottaet al., 2OOG]L. Ballotta and S. Haberinan The fair valuation problem of guaranteed annuit,y options: The stochastic niortality environment case. Insurance: Mathematics and Economics, 38, 2006. [Cocozzaet al., 200Ga,]R. Cocozza and E. Di Lorenzo Solvency of life insura.nce companies: methodological issues. Journal of ilctuarial Practice, 13:81-101, 2006. [Cocozzaet al., 200GblR. Cocozza, E. Di Lorenzo A . Orlando and M. Sibillo In ProThe VaR of the Mathematical Provisions: critical issues. ceedin.gs of the In.tem,ation,a.l Conjeren.ceof Actuni-ies, 2:95-102, 2004. 1ittp://www.papers.ica200G.coiii/Papiers/2002/2002.pdf, 2OOG. [Coppolaet ul., 2002lM. Coppola, E. Di Lorenzo and h l . Sibillo Further Remarks on Risk Sources Measuring in the case of a Life Annuity Portfolio. Journal of Actuarial Practice, 10:229-242, 2002. [Coppolaet al., 2003]M. Coppola, E. Di Lorenzo and M. Sibillo Stochastic a.nalysis in life office managenlent: application to large annuity portfolios. .4pplied Stochustic Models in Business u71d In&ustry, 19:31-42, 2003. [Coppolaet al., 2OOGa]M. Coppola, V. DAniato, E. Di Lorenzo, M. Sibillo Risk measurement and fair valuation in the life insurance field. InEcople: from tradition t,o complexinty, Ca.pri, 2006. [Coppolaet al., 200Fb]M. Coppola, V. DAniato, M. Sibillo Fair value and demograpliic aspects of the insured loan. InProcee&inys of 1 Uth Intel,national Congress on Insurance: Mathematics and Economics Leuven, http://www. kuleuven. be/ime2U06/abstract.php ?zd=80. [Di Lorenzoet al., 2002lE. Di Lorenzo and M. Sihillo Longevit,y risk: measurement and application perspectives. InProceedings of the I 1 Conference in Actuarial Scie71ce, Sarnos, 2002. [Olivieri , 2001lA.M. Olivieri Uncertainty in niortality projections: an actuarial perspective. Insurance: Mathematics and Economics, 29:231-245, 2001. [Olivieriet ak., 2003lA.M. Olivieri and E. Pitacco Solvency requirements for pension annuities. Journal of Pension Economics and Finance, 2: 127-154, 2003. [Orlandoet ul., 20061A. Orlando and M. Politano Further remarks on risk profiles for insurance participating policies. InProceedings of the Ma4F Conference, Salerno, 2006. [Pitacco , 2004lE. Pitacco Sun~icralm.odels in a dgnnmic context: a surtieg. Insurance: Mathematics and Economics, 35(2):279-298,2004,
Location as risk factor Spatial analysis of an insurance data-set Ildikb VitBz' Department of Probability Theory and Statistics Eotvos L o r h d University P b m h y P. stny. 1/C Budapest, Hungary (e-mail: vildikoOcs.elte.hu) Abstract. Our aim was to examine the territorial dependence of risk for household insurances. Besides the classical risk factors such as type of wall, type of building, etc., we consider the location associated to each contract. A Markov random field model seems to be appropriate to describe the spatial effect. Basically there are two ways of fitting the model; we fit a GLM to the counts of claims with the classical risk factors and regarding their effects as fixed we fit the spatial model. Alternatively we can estimate the effects of all covariates (including location) jointly. Although this latter approach may seem to be more accurate, its high complexity and computational demands makes it unfeasible in our case. To overcome the disadvantages of the distinct estimation of the classical and the spatial risk factors proceed as follows: use fist a GLM for the non-spatial covariates, and then fit the spatial model by MCMC. Refit next the GLM with keeping the obtained spatial effect fixed and afterwards refit the spatial model, too. Iterate this procedure several times. We achieve much better fit by performing eight iterations. Keywords: GLM, insurance, Markov Chain Monte Car10 (MCMC), Markov
random field, Spatial statistics.
1
Introduction - Risk models with spatial components
Numerous models are known in the literature for estimating the spatial effect in various problems such as disease mapping, deliquency, or number of accidents [Aratb et al. (2004)], [Gilks, W. R. et al.]. Alongside the spatial effect, however, other influential variables may exist, and they can be quite different depending on the subject of study. For example age and sex are important factors in the case of a motor TPL insurance, while they are insignificant for household insurances. In the latter case type of wall and type of building seem to be important parameters of the contracting partner affecting the number of claims. Besides these more usual (hereinafter: classical) variables location can be an important risk factor worth including into the model. Basically, there are two ways of dealing with the classical and the spatial risk factors; we can estimate the classical ones first, and then fixing their values as if they were known, estimate the spatial effect, or we can estimate all effects simultaneously. Both methods work better than the ones ignoring territorial dependence when the considered phenomenon does contain a 130
Location as Risk Factor
131
spatial effect, but as the usual critic of the first method goes it is not reasonable to cope with the different variables separately, simultaneous estimation is more accurate. Though this latter concept seems to be more desireable but the realisation can be hampered. These models are usually several- level hierarchical Bayes models as spatial effects are generally choosen to be random variables with some prior distribution and using MCMC algorithm for estimating the parameters is often necessary. In this case too many parameters result in cumbersome full conditionals and enormous running time. To avoid this difficulty we suggest the following. Fit first a generalized linear model to the classical risk factors then keeping the received parameters fixed estimate the spatial effect. Return then to the classical risk factors and refit the GLM keeping now the spatial effect fixed, and then move on to the spatial estimation with the newly fixed GLM parameters. By iterating these steps much better predictions can be obtained and the disadvantages of the distinct estimation of the classical and the spatial risk factors can largely be eliminated. The paper is organized as follows. Section 2 describes our model. Section 3 presents the MCMC implementation. The results of the parameter-estimation are presented in Section 4. Section 5 tests the goodness of model choice by comparing the results from our model with ones obtained from some alternatives.
2
Model construction
Selecting the explanatory variables in the generalized linear model we found that: "time spent in risk", "type of building" (4 types), "type of wall" (4 types), "type of roof' (6 types), "type of tarif" (4categories), and "population size of the locality" (10 groups), (all of them are factor-type variables in the GLM) are the significant classical risk factors. As our aim is to analyse the effect of location we include the region of the contract into our model. We build up a hierarchical Bayes model, because this way we can incorporate our prior belief about the structure of the regional effects. At the lowest level of the hierarchy we suppose the counts of claims to comply with the Poisson distribution: yi N P d s ~ m ( t.iEi . A,<) where yi denotes the observed number of claims of client i during the insured period, Ei is the expected number of claims per day based on the non-spatial variables, ti is the number of exposure days i.e. the days spent in risk, Xk = exp(uk), where u k is the relative risk of region k, and ri denotes the index of the region that contract i belongs to. We used a generalized linear model for approximating the effect of the classical risk factors. Using the received parameters we calculated the Ei values of each client:
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Recent Advances in Stochastic Modeling and Data Analysis
where @buizding is the vector of the effects of the different types of building, and bi denotes the type of building belonging to client i. The other notations can be interpreted similarly. The prior distribution of the relative risks of the regions - denoted by {ui, i = 1 , 2 , . . . ,m } , m being the number of regions - is given by a Markov Random Field model. To create a neighbourhoodsystem we regard two regions as neighbours if the distance between their centers is less than 35 kms. The choice of the distance ensures that all regions have at least one neighbour. Would it happen that no claims incur or no data of claims is available in a region without neighbour the estimated relative risk for that region would be zero, which is clearly a nonsense. The same situation arises if we use uncorrelated spatial effects. So, working with correlated regions has the advantage that neighbouring regions influence the relative risk of each other - which is a sensible assumption. In the Markov Random Field model the relative risk of region a is normally distributed with the averaged relative risk of the neighbouring regions as expected value and with variance which is an overall variance parameter u2 divided by the number of neighbours:
where the hyperparameter c7 - denoting the strength of correlation - is a random variable with inverse Gamma(a,X)prior distribution, 61 is the set of the neighbours of region i. The symbol [zly]denotes the conditional density function fx(zlY = y ) in continuous cases, and the conditional probability P ( X = zlY = y ) in discrete cases The joint prior distribution of the parameters 