Correlation and Dependence

G if Vz < *', V 0 < u < 1 Fx.tX{u) > Gx.,x{u).

(3.13)

The equation above implies that the distribution G is more concentrated than the distribution F. We can verify that if F corresponds to the independence situation for G, then FXyX'(u) = u, and we retrieve the proposition given in Section 3.2.5 for PRD. Note that this ordering is not symmetric in x and y and consequently we cannot verify the axiom of exchangeability. 3.6.4

Association

ordering

The notion of association of variables prompted Schriever [193] to introduce the following order <£.ass '• Definition: Let (X, Y) and (W, Z) be two random bivariate vectors having the distributions F and G respectively. Then :

F
Ki(xi,yi)

< K1{x2,y2)

,K2{xi,yi)

> K2(x2,y2)

-> x1 < x2 ,y x > y2 . (3.14)

This new condition is required to assure that the <£ass ordering will be applicable when the variables (X, Y) are not exchangeable. Properties of this ordering are: - Association-ordering is invariant under scale and location transformations.

Positive Dependence

57

Orderings

- PRD-ordering implies association-ordering. Namely, if F <^PRD G, then it suffices to take Ki(x,y) = x to arrive at F <£ass G. Example: Let X and Y be two random variables and define Xa = ( 1 - aX) + aY, Ya = aX + (1 - a)Y" with a 6 [0,1/2], then : V0
1/2 , {Xa\Yai)

« a s s (XQ2,ya2).

This example has been used by several authors, see e.g. Schriever [193] and Yanagimoto [226]. 3.6.5

PDD-ordering

Let (X, Y), (W, Z), be two random vectors with distribution function F and G. Utilizing the concept of positive function dependence (Section 3.2.4), Rinott and Pollack [182] introduce the ordering

if Cov{h(X), h(Y)) <

Cov{h{W),h{Z))

for every real function h. Since the two distributions have the same marginals, this is equivalent to E{h{X)h{Y))

<

E(h{W)h{Z))

for every real function h. It is then possible to extend this ordering to the case of two random vectors X = (Xi,X2,...,Xn) and Y = (Yi,Y2, ...,Yn) (n > 2) : X « P D D Y if E{Wl=lh{Xi))

<

E(Un=1h(Yi))

Rinott and Pollack established the following characteristic property: F

<£.PDD

G <£> F(x, y) — G(x, y) is a positive definite kernel.

The PDD-ordering is preserved under mixture and under limits in distribution. The preservation under mixture means that: if (X, Y, d) and (W, Z, 6) are two random vectors depending on a parameter 6 £ 0 , then

ye e G, [(x, y)/© = e] (x, Y) « P D D (W, Z) .

58

Concepts of Dependence and Stochastic

3.6.6

Orderings

defined from DTP(0,1)

Ordering

and

LED

In the same manner as in Section 3.6.3, we can generalize the inequality (3.10), which compares the ratio s*, ("\ with the ratio obtained (under independence) from the uniform survival function jE%, for comparing two distributions as follows : F « O T P(o,i) G

ifVx<x'V0
f§*£l < f£*M b

x'x\V)

^x'x\V)

(where the exponent in the survival function indicates the associated distribution function). To define the DTP(0,l)-ordering, Caperaa and Genest require additionally that : Gx'X{u) Gx'X(v) ~

Fx:x(u) Fx:x(v)' SI

Equivalently, one can assert that the ratio -p^-(u) with x < x' is an " x1 x

increasing function of u. The LRD-dependence is expressed as : F<€.LRDG

ifVKi', V0x(u)'

.

If we suppose that the c.d.f Fx<x and Gx<x have densities, we can rewrite this last inequality as dGX'X(u) dFx,x{u) is an increasing function of u, when x < x'. The preceding generalization is due to Caperaa and Genest in [37]. Kimeldorf and Sampson [129] have defined another TP2-ordering ( < T P 2 ) as follows : Let 7 x J be a rectangle, and F(I, J), G(I, J) be the associated probabilities . We write h < I2, if Vz £ h and \/y G I2, x < y. F «TF2 G F(I1,J1)F(I2,J2)G(h,J2)G(I2,

if

VJi < h, VJi < J 2 ,

JO <

F(I1,J2)F(I2,J1)G(I1,J1)G(I2,J2). (3.17)

Positive Dependence

59

Orderings

This ordering is different from the LRD-ordering suggested by Caperaa and Genest [37] and does not imply the DTP(0,1)- and the PRD-ordering as the LRD-ordering does. (Caperaa and Genest provide a counterexample to show this non-implication.) Proposition: Let F and G be two distribution functions, with the same marginals and such that the conditional distribution FY/X and GY/X have supports independent of x, then : F <€LRD G => F • F - C P R D G.

Examples: (1) Yanagimoto and Okamoto's example [227] Let X = U,Ya = (l + aU)V with a > - 1 and U and V be independent random variables with an absolutely continuous distribution. Yanagimoto and Okamoto have shown that this family is ordered by PRD-ordering, and Caperaa and Genest [37] have proved that it is ordered by LRD-ordering. (2) Frank's Copula (see chapter 4, Section 4.6.2) The copula is defined by: Fa{x,y)=loga{l+K-

(ax - l)(ay - \\ ^ tl) a — 1

0< a < 1

and for a = 1, Fi(x,y) = xy for 0 < x,y < 1 This family possesses the LRD-ordering , namely one can verify that for x < x', a' < a the ratio Ja,x',x V^j fa',x',x{u)

is increasing with u, which is the characteristic property of LRDordering defined above (relation 3.16). (3) Clayton's Copula (see Chapter 4, Section 4.6.2) Here the c.d.f is Fa{x,y)

= {x-a+y-a-l)-1/a,

Denote Fa<x>tX(u) = FatX'F~l(u)

a>0,0<x,y
,

(u)

a x ordered by the DTP(0,l)-ordering, namely the ratio &''*'',!, with "a,x' ,x \ ) a < a', is not an increasing function of u. u

60

Concepts of Dependence and Stochastic

3.6.7

Integral stochastic

Ordering

orderings

In this section we do not impose that the two vectors to be compared have the same marginals. Consider a random vector X in Rn for n > 2. The orderings to be defined are called integral stochastic orderings, because they are defined for a class of real-valued functions T, such that X « Y holds if and only if E{f(X)) < E(f(Y)) for all / belonging to T (for which the expectations exist). For example J- can be the set of all component-wise increasing functions, or the set of all convex functions , or all supermodular , or all componentwise convex, or all directionally convex functions. (A function is directionally convex if it is supermodular and componentwise convex.) We have already seen in Section 3.2.2 that PQD-dependence is closely related to certain properties on supermodular and componentwise increasing functions. We shall give here a few properties of supermodular ordering, which show that this ordering is suitable for the comparison of distributions with the same marginals. Later we discuss directionally convex ordering as an ordering for vectors with different variability in the marginals.

3.6.7.1

Supermodular ordering

(1) X « s m Y implies P ( X > x) < P ( Y > y) and P ( X < x) < •P(Y < y). Thus supermodular-ordering implies upper-orthantordering and lower-orthant-ordering (where upper and lower orthant orderings are natural extensions of PQD-ordering). (2) X -C s m Y implies Xi = Yi for all i — 1,..., n. Therefore the supermodular ordering presupposes that the two vectors have the same univariate marginals and therefore compares only their dependence structure (i.e. their copulas, see Chapter 4). (3) X « s m Y implies cov(f(Xi),g(Xj)) < cov(f(Yi),g(Yj)), for all / and g R M- R both increasing or decreasing and for all i and j in l,2,...,n. (4) The Lorentz inequality. Let (X\,X2,- • • ,Xn) be identically distributed random variables. Then F
Positive Dependence

Orderings

61

as X, then X* -CSm X . 3.6.7.2

Directionally convex ordering

Sometimes it is of interest to compare some positive convex combinations of the components of the two random vectors X and Y. If the two random vectors have a common copula, then if their components are ordered by the increasing-ordering (Eg(Xi) -C Eg{Yi)) for all i and for all g increasing functions), this is also true for the vectors (see Scarsini [190]). But if the components are ordered just by the convex order (g is convex), then this is not true for the vectors [190]. The idea of Miiller and Scarsini [159] is then to use the directionally convex functions which combine the properties of convexity of the components and the supermodularity property of the function to construct an ordering. They show the following two properties: - If X and Y have a common copula which is CI (conditionally increasing) and if Eg(Xi) < Eg(Yi) for all convex-functions g : R >-» R, and all i = 1,2, ...,n then Ef(X.) < Ef(Y) for all directionally convex functions / : Rn H-> R. - If Efi(Xi) < Efi(Yi) for all convex functions fi implies Ef(X.) < Ef(Y) for all comonotone random vectors X and Y, then / is directionally convex. (A vector X is comonotone if its distribution is the upper Frechet bound.)

3.6.8

Generating

a family

of partial

orderings

Yanagimoto [228] followed by Metry and Sampson [155] provided an approach to unify the TP2-ordering and the concordance ordering as well as to generate other orderings. The idea is to adjust the endpoints of the intervals of the rectangles h x Ji and J2 x J 2 utilized in the definition of the TP2-ordering. An endpoint of any interval can be ±oo, and the two intervals Ix and I2 (or Ji and J2) may or may not possess a non-empty intersection. In this manner Metry and Sampson generate 64 orderings, of which concordance ordering and TP2-ordering are particular cases. More precisely, with the following notations : - L is the set of left intervals : ] — 00, a[ or ] — 00, a] - R is the set of right intervals :[a, +oo[ or ]a, +oo[ - S is the set of any arbitrary intervals (L C S and R C S)

62

Concepts of Dependence and Stochastic

Ordering

- If two disjoint intervals share a common endpoint Ii < h, the corresponding notation will be A (A stands for "abut"), and NA if they (not necessarily) share a common endpoint (the two disjoint intervals are not required to "abut", but may do so) . For example the concordance ordering is represented by (L,R; L,R; A, A), and the TP 2 -ordering by (S, S; S, S; NA.NA). (S,S;S,S;A,A) corresponds to an ordering defined as P(3,3)ordering by Yanagimoto [228]. Thus, using this notation, we have :

(.,.;.,.;NA,A) / (.,.;;,.; N A, N A)

\ (.,;.,; A, A)

\

s (.,;.,; A, NA)

where any combination of L, R and S may be substituted for the first pair ".,.". (The two arrows /• and \ mean "implies".) Moreover, (S,R;.,.;.,.) / (5,5;;,.;.,.)

\ (L,R;.,.;.,.)

\

/ (L, 5;.,;.,.)

where any combination of L, R and S may be substituted for the first pair ".,." and any combination of A and NA for the second pair ".,.". All these orderings satisfy the conditions P1-P6, and P9 of Kimeldorf and Sampson [130]. The property P7 is satisfied only for intervals containing positive and negative values for the first variable, that is the 32 orderings (L,R;.,.,;.,.) and (S,S;.,.,;.,). The property P8 is satisfied only by the orderings for which the intervals for X and Y can be interchanged, that is (L,R; L,R;.,.) and (L,R; L,R;.,.), (S,R; S,R;.,.) and (S,S;S,S;.,.). Among these orderings some of them are equivalent: ( 5 , 5 ; 5 , 5 ; NA, NA) ( 5 , 5 ; 5 , 5 ; A, A) that is TP2-ordering is the same as P(3,3)-ordering. Also, (L, R; L, R; A, A) = (L,R;L,R : NA,NA). The ordering on the left of the equality sign was defined by Tchen [216]. It is of interest that

Bayesian Approach to Stochastic

Dependence

63

the concordance ordering on the left of the equality is equivalent to a more general ordering on the right. This is perhaps an indication of the difficulty inherent in defining fully consistent orderings.

3.7

Bayesian Approach to Stochastic Dependence

Inspired by Bayesian methodology, Brady and Singpurwalla [31] argue that the notion of dependence or independence between two or more variables is conditional on a known or an unknown parameter 9 or (latent) variable. For example if (X, Y) is a pair of normal variables, then they are independent or dependent conditionally to their correlation coefficient p. Thus if we can define a prior distribution (P) on the parameter p, we shall be able to associate a certain probability for independence or positive dependence of the pair (X,Y). Definition 1 : Suppose that the two variables (X, Y) are independent or quadrant dependent according to the values of a parameter ©. More precisely : (1) F(x,y/e (2) F(x,y/Q (3) F(x,y/Q

G h) = Fx{x/Q G h).F2(y/e 6 I2) > F^x/e G I2).F2(y/e G h) < F^x/Q G h).F2(y/Q

G h), V(z,y) G I2), V(x,y) G / 3 ) , V(x,y)

where 7j, i = 1,2,3 is a partition of R. Then the two variables are independent, positive quadrant dependent, or negative quadrant dependent conditionally on 0 in I\, I2 or I3. Definition 2: (1) P{{X,Y) independent) = P ( 0 G h) (2) P((X,Y)PQD) = P(QeI2) (3) P((X,Y)NQD)

=

P(Oeh)

If conditions 1, 2 and 3 are fulfilled and the expectations exist, then : (1) E(X,Y/Q (2) E{X,Y/Q (3) E(X,Y/Q Definition 3 :

G h) = E(X/Q G h).E{Y/Q G h) G I2) > E{X/Q G I2).E(Y/Q G J2) G /3) < E(X/Q G h).E{Y/Q G / 3 ) .

64

Concepts of Dependence and Stochastic

Ordering

If p is the correlation coefficient between X and Y, and if a prior distribution on p is defined, we can compute the probability : U(a)=P(\p(X,Y)\>a) which is termed by Brady and Singpurwalla a "correlation survival function ". Definition 4: X and Y are highly stochastically correlated if P(\p(X,Y)\>a)>l-a for 0 < a < 1. Definition 5 : The pair (X, Y) is stochastically more correlated than the pair (X',Y') if P(\p(X,Y)\>a)>P(\p(X',Y')\>a). Definition 6 : The pair (X, Y) is stochastically more correlated in expectation than the pair (X',Y') if / IIx,y(Q)dci > / ILx',Y'(oi)da where fnx,Y{a)da = P(\p(X,Y)\ > a). Orderings of random variables seem to be a fruitful and unexhaustable area of research which attracts both theoretical and applied statisticians, often motivated by newly arising real-world situations.

Chapter 4

Copulas

4.1

Introduction

A copula provides a uniform representation of a bivariate distribution F on the unit square. This result is due to Sklar (1959) [208], and has been revisited by Cartley and Taylor (Section 4.1). Copulas are an important part in the study of dependence between two variables, since they allow us to separate the effect of dependence from effects of the marginals distributions. This feature is analogous to the bivariate normal distribution, where the mean vectors are unlinked from the covariance matrix and jointly determines the distribution. Many authors (Gumbel, Plackett, Mardia, Ali, Mikhail and Haq, Clayton, Joe, Genest, Nelsen - some of them cited in the references) have studied constructions of bivariate distributions with given marginals : this can always be viewed as constructing a copula (Section 4.4). Starting from any real integrable function on the unit square, Riischendorf (Section 4.5) has proposed a general method for construction of copulas. This method allows us to generate, for example, all the polynomial copulas (Section 4.5.2). Long and Krzysztofowicz utilized a particular case of this method. Another manner to generate copulas is a mixing with respect to a third random variable (called frailty variable). Marshall and Olkin have systematized this idea (Section 4.5.6). Other authors have studied characteristic properties of sub-family of copulas. Genest et al. and Sungur et al. have considered particularly Archimedean copulas (Section 4.6), Caperaa et al. have constructed a more general family of Archimax copulas (Section 4.7). Wei studied copulas with discontinuities (Section 4.8). It is often possible to generalize the method of constructing copulas 65

66

Copulas

for more than two variables, but there are compatibility constraints on the marginals. Chakak and Koehler and later Sungur studied a particular case of m-copulas with a property of truncation invariance (Section 4.9.2). A few simulations procedures can be derived from the properties of copulas (Section 4.4.10). The purpose of this chapter is to present - in an organized manner- numerous results on copulas scattered in diverse literature with the emphasis on dependence concepts and properties. Some results are unified. 4.2

Definition a n d Some P r o p e r t i e s

It is well known and easily verified that F\(X) and F2(Y), where F\ and F2 are the marginals distributions of X and Y respectively, are two uniform variables if F\ and F2 are continuous. Hence if the marginals Fi and F2 of the bivariate distribution F are continuous, there exists a unique copula, which is a cumulative distribution function, with its marginals being uniform. Formally a function C : [0, l ] 2 —> [0,1] such that F(x,y) = C(F1(x),F2(y))

(4.1)

is a copula. On the other hand, if C(ui,U2) and continuous F\ and F2 are given, then there exists an F such that: F{F^\u1),F2\u2))

=

C{uuu2)

Fi(t), i — 1,2 is continuous and non decreasing, but could be constant on some intervals. In that case, one defines a quasi-inverse by Fr1{t) = inf{x

:Fi(x)>t).

Using copulas allows us to separate the study of dependence from the study of the marginals, since one is then reduced to study of the relation between two uniform variables. If the density /(.,.) of F(.,.) exists, one can derive from (4.1), the relationship between the density / of F and c of C: f(x,y)

= c(F1(x),F2(y))f1(x)f2(y)

where fi{x) and f2(y) are the marginal densities of F. (1) Elementary properties

,

(4.2)

Definition

and Some

67

Properties

C(u, 1) = P{U < u, V < 1) = P(U < u) = u and similarly C(l,v)

(4.3)

= v. Also

C(u, 0) = P(U < u, V < 0) = 0 .

(4.4)

(2) Rectangular inequality Since C(u,v) is a distribution function, it satisfies for all 0 < Ui < u2
- C(Ul,v2)

- C(u2, vx) + C(ui, wi) > 0.

This inequality implies in particular that C(u,v) is increasing in both variables. (3) Continuity A copula is continuous in u and v; actually it satisfies the stronger Lipschitz ([196]) condition: \C{u2,v2) - C ( u i , t > i ) | < \u2 - u i | + \v2 -vi\.

(4.5)

(4) Differentiability Since C(u,v) is increasing and continuous in the two variables, it is differentiable almost everywhere and from (4.5) we see that 0 < &C(u,v) < 1 and 0 < &C(u,v) < 1. (5) The survival function of a copula Using (4.3) the survival function C corresponding to C(u, v) is : C(u, v) = I — u — v + C(u, v). From (4.3) and (4.4), we have : C(u, 1) = 0 C(u, 0) = 1 . Starting from the pair (1 — U,l — V) we can define another copula C'(u, v) whose survival function is connected with C. Namely, given C'(u,v)

=

Pr{l-U


< v) =

68

Copulas

C(l-u,l-w)

=

u + v - l + C(l-u,l-v)

(4.6)

we then have : Cr{u,v)

=C(l-u,l-v).

(6) The dual of a copula The dual of C is the function d defined by Cd{u,v) = u + v — C(u, v). Note that Cd is not a copula since it does not satisfy the equation (4.4):C d (u,0) is equal to u, not zero. However it can easily be verified that 0 < Cd(u,v) < 1. (7) A diagonal copula: A diagonal copula is the single variable function defined by D(u) = C(u,u). We can also define the dual of a diagonal copula Dd(u) = 2u — D(u). If U and V are independent, let C° be the corresponding copula. Then C°(u,v) = uv and D°(u) = u2. A diagonal copula can also be viewed as the distribution function of W = max(U, V). Indeed : Fw{w) = Pr{max{U,V) Pr{U <w,V

<w) =

<w) = C{w,w)

=D(w).

In the same manner, the dual of a diagonal copula Dd{w') is the distribution function of W = min(U, V).

4.3 4.3.1

The Frechet Bounds Lower and upper Frechet bounds in the family T{Fi, of bivariate distributions with common marginals

F2)

Let J-(Fi,F2) be the family of the bivariate distributions with common marginals F± and F^. This family has two limits which can be attained. Indeed, since F(x,y) is a non-decreasing function in x and y, then: F(x,y)
= F1(x)

The Frechet

Bounds

69

and F(x,y)

< F(oo,y) = F2(y)

i.e., F(x,y)<min{F1(x),F2(y)). In the same manner, the survival function S verifies: S(x,y) F+(x,y) = min(Fi(x),F2(y)) On the other hand

<

min{Si(x)S2(y))

is called the upper Frechet bound.

F(x,y) = S(x,y) - 1 + F^x) + F2(y).

(4.7)

However S(x, y) is always positive or zero, thus: F{x,y) > max(0,F1(x)+F2(y)

- 1)

F~{x,y) = max(Q,Fi(x) + F2(y) — 1) is called the lower Frechet bound. Similar relations exist between S, Si and S2. 4.3.2

The Frechet bounds for a copula

The supports of the copulas of the Frchet bounds are the diagonals:the main diagonal to the upper bound, and the other to the lower one. We shall show this result for the upper Frechet bound C+(u,v) : C+(u,v)

= =

min(u,v) w-l[o,t,](") +v.l]„,i](u)

(4.8)

where l[0it)](u) is the indicator function of [0,v]. Let v be fixed. The derivative with respect to u (with an extended definition of the derivative, see L. Schwartz [195]) is DuC+(u,v)

=

l[0,v](u) +u(-5v(u))

=

l[o,v](u) +

=

l[o,„](«),

+v(-6v(u))

(v-u).Sv(u) (4-9)

70

Copulas

where 6v(u) is the Dirac measure on u = v. One can write l[0,„](u) = l[u,i]( u ) and 5v(u) = Su(v). Hence the derivative of g(v) — DuF*(u,v) with respect to v is DvDuC+(u,v)Dv

= 8u(v) .

Therefore, for the upper Prechet bound of the copula, the measure is concentrated on Su(v) , which is the main diagonal (£>i : u = v). Similarly C~(u, v) = max(0,u + v — 1) has its probability mass concentrated on the second diagonal. In the general case, the upper Prechet bound is the "curve" (Fi(x) = F2{y)). One can easily verify that the limits of the dual of a copula and of the diagonals copulas are: min(u,v)

< Cd{u,v) < 1 ,

max(0,2u — 1) < D(u) < u , and u < Dd(u) < 1 . Using the diagonals of a copula, we can write a condition for the maximal (positive or negative) dependence : If C = C+, then Fmax(uy)(w) = Fmin(u,v)(w) = D+(w) = D+(w) = w, if C = C~, then Fmax(Uy)(w) = D~(w) — max(2w — 1,0) and Fmin(u,v){w) = D^(w) = min(l,2w). 4.4

Examples (1) The independent copula : C°(u,v) = uv is the copula associated with a pair (U, V) of independent variables. (2) A linear convex combination of copulas is a copula. For example: C{u,v) =a.C+(u,v)

+ (l-a).C-{u,v)

0 < a < 1.

(3) The family of Farlie-Gumbel-Morgenstern (FGM) copulas are : C a (u,w) = u t ; [ l - l - a ( l - « ) ( l - v ) ]

|a| < 1.

(4.10)

This family does not contain the upper and lower Frechet bounds.

71

Examples

(4) The Iterated FGM copulas The survival function of an FGM-copula is Ca(u,v)

= 1 - u — v + Ca(u,v)

— (1 — w)(l — v)[l + a.uv]

Kotz and Johnson [132] "iterate" the FGM copula by replacing the term (1 — u)(l — v) by the survival function Cai(u,v) for some |«i| < 1 to obtain : Caua2(u,v)

=

uv[l +

a2Cai(u,v)]

=

uv[l + a 2 ( l - u)(l - v)(l + axuv)] . (4.11)

Lin used the same idea and "iterated" the FGM copulas by replacing the term uv by Cai(u,v) in the formula 4.10 and obtained: £ / a i , a 2 ( W ) v)

=

uv[l + a 2 ( l - w)(l - v)(l + a i . ( l - u)(l - v)}.

(4.12)

See Chapter 5 for additional details of FGM copulas and their generalizations. (5) Copula associated with the Bivariate Normal Distribution (BVN) Let (X, Y) be a pair of random variable, with means zero, variances one and bivariate normal distribution: ./V((0,0),£) with :

-(JOLet 4> and $ be respectively the density and the cumulative distribution function of the iV(0,1) distribution and p be the density of the BVN distribution. Let U = $(X) and V = $(Y) be two uniform variables with density c and cumulative distribution function C. Using (4.2), we have: r(v

*• '

V)

;

= 92C(u,v)

dudv

=

^($-'(U),$-'(D))

{$-\u))<j>{$-^(v)) '

i.e.

•<•••» "7IT7-«ri-5(i^ i J ( *" ,( " )) ' + ($- 1 («)) 2 + P$"1(w)$"1(v)] •

Copulas

(6) The Plackett copula: Plackett [169] proposed to construct a bivariate distribution for which the ratio P(U < u, V < v)P{U > u, V > v) P(U > u, V < v)P(U < u, V > v) is constant. This idea generalizes to bivariate distributions with continuous marginals the concept of odds-ratio used for testing independence in contingency tables. This ratio can also be written as :

C(u,v).C(u,v) {u-C(u,v)){v-C(u,v))

'

If this ratio is 1, C = C°, i.e. as in contingency tables, the two variables U and V are independent, if the ratio is a constant 8 different from one, then C(u, v) is a solution of a quadratic equation: „.

,

C{U V)

'

1 + (0 - l)(u + v)± y/{l + (0- l)(u + v)Y - 4uv6(d - 1)

=

2(^1) (4

If we require that 0 < 0 and 6^1 (i.e. a positive dependence), the solution with the sign + is not appropriate for a copula because in that case C(u, 0) is not zero. The solution with sign — is appropriate (C(u,0) = 0 and C(u, 1) = u and its density is positive). This family includes its upper Prechet bound : if 6 goes to infinity, then C tends to C+. Hence this family, with only one parameter, is useful for fitting of empirical data. (7) The Ali-Mikhail-Haq family of distributions 111)

C

' ^ =

1-0(1

-„)(!-«)

I"*1"

If 0 = 0, then we have the independence Cg — C°. This family does not contain the Prechet bounds.

Construction

4.5 4.5.1

73

of a Copula

Construction of a Copula The Riischendorf

method

To obtain a general form for a copula, one can use a powerful and elegant method developed by Riischendorf [186] : let / 1 ( u , v ) has its integral to be zero on the unit square, as well as its two marginals integrate to zero, i.e. : f1(u,v)dudv

/

=0

(4.14)

2

J[o,i]

and / f1(u,v)du Jo

= Oand I f1(u,v)dv Jo

= 0.

(4.15)

(4.15) implies (4.14). In that case 1 + ^{u^) is a density of a copula. l There is however a constraint, namely that 1 + f {u,v) must be positive. If it is not the case, but Z 1 is bounded, one can then find a constant a such that 1 + a.f1 is positive. It is easy to construct a function f1 : one starts with an arbitrary real integrable function / on the unit square with its marginals being uniform, and computes: V = /

•Ao.i] 2

/i(«)

= / Jo

f(u,v)dudv

,

f(u,v)dv

and h{v) = / f(u,v)du ./o

,

then f1 = f - h ~ h + V. If we have two functions Z1 and g1, possessing the properties stipulated above, then 1 + f1 + g1 is the density of a copula, and more generally 1 + S " = 1 / / is a density with / / satisfying (4.14) and (4.15).

74

Copulas

4.5.2

Application

to polynomial

copulas

To obtain a polynomial copula one starts with / = ukvq and obtains f\u,v) = (uk - ~ ) K - ~ )

,k>l,q>l.

Therefore the function : 1 + ^ - ^ K - ^ )

(4.16)

with the constraint : 0<0<min>'(k

+1 +

^ V

(* + D(9 + D

is the density of a polynomial copula. Combining the expressions (4.16) for all k, k > 1, and all q, q > 1, we arrive at the general formula:

££ = !+ £ M«-- F L)K^) fc>l,g>l

(«7)

*

with the constraints :

0<mm( £

9

*fc

E V(

fc 1} (fc + 1 ) ) < l

and, after some simplifications, a polynomial copula of power m can be written as k+q<m — 2

C(«,i,) = «,[! +

£

« (fc + 1 )

^+1)(U*-l)(i,«-l)1.

(4.18)

If one requires, for example, a polynomial copula of the fifth power, its density is written as : d2

°

i + eu{u-\){v-\)

+ ex2{u-\){v2-\)

+ e21{u2-\){v-y v

v v x dudv 2JX 2 ' 2,x 3' The polynomial copula of the fifth power then becomes:

3

u

2'

C(u,v)=uv[l + 0-f(u-l)(v-l) + e-f(u-l)(v2-l) + e-f(u2-l)(v~l)}. (4.19)

Construction of a Copula

75

Using this method, we can generate all polynomial copulas. Indeed the dimension of the space of polynomial copulas of power m is equal to the number of the parameters 9kq'- k varies from 1 to (m — 3), the range of q is the same, but q + k < m — 2, thus number of parameters is then ^ m - 3 i _ (m-z){m-2) ^ T h i g d i m e n s i o n c o i n c i d e s with the one found by Wei et al. [223], using a different argument for polynomial copulas of power m. Expanding the expression (4.19), we can verify that it coincides with the expression furnished by Wei et al. [223] using a different approach. 4.5.2.1

Approximation of a copula by a polynomial copula

If a copula, indexed by a parameter 8, Cg (u, v) belongs to the C" functions with respect to 0 (the functions whose n-th derivative is continuous), we can then express Cg(u, v) by means of the Taylor expansion in the neighborhood of0 o :

^

C^(u,v){6 - BoY e K

Ce(u,v) « C8o(u,v) + £

°

£

°' .

Choosing the value #o which yields the independence i.e. such that Cg0 (u, v) = uv, and if the successive derivatives of C$ with respect to 9 are powers in uv, we then obtain an approximation of Cg by means of a polynomial copula. Examples: (1) The FGM family: The FGM family corresponds to its first order expansion in Taylor's series around 6 = 0 (2) The Ali-Mikhail-Haq family :

Cs{ U

- ^ = x-e(iU-u)\-v)

=

^[i+EW1-")^-^]

1*1 < 1-

(4.20) If we consider in (4.20) the first-order approximation only, we then have the FGM family, and with the second order approximation we arrive at the iterated FGM family of Lin. For an approximation of any order we have a polynomial copula. (3) The Plackett family: Nelsen ([161], page 83) proved that FGM family is a first order approximation to the Plackett family, by expanding it in Taylor's series around 0 = 1.

76

Copulas

(4) The copula associated to BVN family: The first order approximation around p — 0 is Ct>(u,v)=uv[l 4.5.3

Other

+ p*-1{u)*-1{v)]

.

examples

Riischendorf [186] gives two other examples of constructing copulas : (1) A distribution concentrated in the vicinity of the diagonal : Let f(u,v)

=

V\u~v\

The function f(u,v) is not a density of a copula, but one can compute f1 - f - / i - f2 + V where /i(u) = 2 v / w + 2 v / l - u with f2(v) obtained by interchanging u and v, and

V = J /1(«)du=|. Hence: 1 l H - / 1 ^ ) = — ^ = = - 2
Let /«(«,«) = - V . „ l > *

O<0<1

where l| u _„|>0 = 1 when |u — v\ > 6 and 0 otherwise. Then the first marginal is f16{u) = (1 - 26)lei-« for 0 < 0 < 0.5 and /ie(w) = (u - 0)l u >e + {0 -

u)lu
for 0.5 < 6 < 1.

Construction

77

of a Copula

Interchanging u by v in the above expression, we have a formula for the other marginal f2e- Finally the associated density is 1 + fl{u,v)

= 1 + fe(u,v)

- he{u) - f2e(v) + 6(1 - 6).

If 6 = 1, we have the independence case. Our third example is the one proposed by Long and Krzysztofowicz [146], who may not have been aware of the work of Riischendorf. (3) Copula with scalar parameter and a power function characteristic: Long and Krzysztofowicz [146] express the density function of a copula as f(u,v)

= 1 + 9c(u,v)

with the same constraints on 8 (a scaling parameter) and c(u,v) as Riischendorf. Next using the ideas of Kotz and Seeger [132] related to the methods of constructing densities of a copula (termed in the paper of Kotz and Seeger "a density weighting function"), they specify the function c(u, v) in such a manner that it possesses symmetries about the main and second diagonals, and in case of positive dependence it has a ridge along the main diagonal (the upper Frechet bound) and a through along the second diagonal (the lower Frechet bound). The structure is opposite in the case of negative dependence. More precisely, they utilize a function K, called the regression characteristic to define c(u,v). K is defined on [0,1] by K(£) = (1 - t)@ or n(t) = t0 for /? > 0. Now : c(u,v) = ci(u,v) +c2(u,v)

- 2K(1)

where the function K(u) = /Qu K,(t)dt, and ci(u,v) = K(\V — u\) , v f c2(u,v) = i [

K[U + V)

\

K(2

' — u — v)

for u < 1 — v tor u > v.

If n(t) is decreasing and 6 > 0, then the pair (£/, V) has a positive dependence with a ridge along the main diagonal due to ci(u,v) and a through along the second diagonal due to c2(u,v), and a negative dependence if 9 < 0. Conversely, if K(<) is increasing and

78

Copulas

6 > 0, then the pair has negative dependence with a ridge along the second diagonal and a through along the main one. Parameters 6 and /? are interconnected. Namely K(l) = 4 r and 0 < K,(t) < 1, imply that -gq^- < c(u,v) < 1 and since the density 1 + 9c(u, v) is positive we have :

-i^L<6 1. On the other hand, if n(t) = t@, then the correlation coefficient is between — 1 and 7/12. Therefore it is possible to obtain all the degrees of dependence between U and V using this model. 4.5.4

Models defined from a distortion

function

In the field of insurance pricing (see e.g. Frees and Valdez [75]) one uses a distortion function g which satisfies g : [0,1] i-> [0,1], with g(0) = 0, g(l) = 1 and g is increasing. Starting with the distribution function F(x,y) = C[Fi(x),F2(y)], one defines another distribution function F(x,y) = g[F(x,y)} with marginals g(Fi) and g{F2). The associated copula is then C(u,v) = g[C(p - 1 (u),3 - 1 (v))]. For example with an exponential distortion function: 1-

eat

*W = T37^'

a

>°

and with an independent copula C(u, v) = uv, we find : C (u, v) = log a (l + *

^'-) a —1 which is the well known Frank's copula (Frank, 1979 [72]). 4.5.5

Frailty

models

The frailty models have been defined and widely used in the field of survival analysis (see, for example, Kalbfleisch and Prentice [122] page 33, Hougaard [100], Oakes [163]). Recall that h(x) = -£lnS(x) is the marginal hazard

Construction

79

of a Copula

of X, H(x) = f£ h{u)du — -lnS(x) the cumulative hazard of X. In these models, a frailty variable W, with distribution function G(w) acts multiplicatively on the marginal hazards. For example for the random variable X: hi(x/w)

= hio(x).w,

where h\o(x) is the baseline hazard. Noting that Sio(x) — exp(—Hio(x)) we obtain Si(x/w)

= exp(— / hi0(u)du.w) Jo

= exp(-H10(x)w)

= (Sio{x))w .

Therefore the marginal Si (x) is 5i(;r) = f exp(-Hio(x)w).dG(w)

=

fS?0{x).dG(w).

Furthermore, one supposes that, conditionally to W, the variables X and Y are independent. Hence, we have : S((x,y)/w)

= Si(x/w).S2(y/w)

= exp[—w(H10{x) + H2o(y))] ,

and integrating: S(x,y)

= Jexp(-w[H10{x)

+ H20(y)])dG{w)

.

(4.21)

Finally S(x,y)

=

E(exp(-wH0))

with H0(x,y) — Hi0(x) + H2o{y)- Thus S(x,y) is defined on ]0, oo[2 with the Laplace transform y _ 1 (the reason for this designation will be clarified in the next section). But, since Si =
80

Copulas

4.5.6

Marshall

and Olkin's

generalization

In frailty models, the survival function S is obtained from the marginal survival functions Si, i = 1,2 by Laplace transform. This procedure can also be applied to F,, i = 1,2 to obtain another distribution function F : F(x,y) =

1

V-

MF1(x))+lp(F2(y))].

More generally, let G(wi, w2) be a mixture distribution denned on ]0,00[2 with the Laplace transform
F{xx,x2)

= J JK(F^(x1),F^)(x2)dG(w1,w2).

(4.22)

A straightforward generalization for the case of n (n > 2) variables yields : F(x1,..,xn)

= J J K(F^(x1),...,F^(xn))dG(w1,...,wn).

(4.23)

Various G and K functions result in a variety of multivariate distributions functions (or survival functions) denned by means of their marginals. For example: (1) K is the independent bivariate distribution and G is the upper Frechet bound (UFB): In that case G{wi,w2) = mim(Gi(wi)), and we retrieve the frailty model defined in the preceding section: F(xi,x2)=
+
which can be generalized to case of n (n > 2) variables :

F(xu...,xn) = ip-^MKixi))]

•

(4-24)

Various examples with the Gamma and stable Laplace tansforms are provided in the next section (Archimedean copulas). In that case, the function F(xi,...,xn) is TP^ (totally positive of order

Construction

of a Copula

81

infinity in each pair of arguments), and in particular it is TP2 (see Chapter 3, section 3.2.8). If G has a negative binomial distribution with the Laplace transform ip~l(t) = (\Pge-t)

> a > 0 , 0 < p < l , q - l - p , and the

inverse function
ta

±) then

p+qta

F(xi,x2)

=

f—-—-——1

•

(4-25)

[l-q(l-F1-(x1))(l-F2-(x2))}The Ali-Mikhail-Haq distribution is a particular case where a = 1, with a more relaxed definition of — 1 < q < 1. (2) K is an independent bivariate distribution, G has convolution form: We shall assume here that the frailty variables are W\ = U + W and W2 = V + W, where U, V, W are independent. Then the Laplace transform of the pair (Wi, W2) is

,

In this case F(x1,x2)

= ipi[2[q2F2(x2)].ipol
Specifically, If l^qe-t)

« i > 0 , * = 1,2,3,

a10 = ai + a0 > 0 ,a20 = a2 + a0 > 0, then qi(t) = —ln(

—.

)

^l/aio

92(<) = -ln(

—,

)

p + qt1/"™'

Copulas

and ^(11)^2(12)

F{xux2)

[1 - q(l - F^° ( n ) ( l - F2a™ ( 1 2 ) ) ^ ) ] ° " This is a generalization of the family (4.25) obtained with a negative binomial Laplace transform. Another application is the correlated gamma frailty model of Yashin and Iachine [230], written for the survival function and where U, V, and W have gamma distributions, U and V with mean 1 — p and variance (1 — p)8, and W with mean p and variance p6. The resulting variables W\ and W2 have gamma distribution with mean 1 and variance 6, and correlation coefficient p. The joint survival function is S(xux2)

= S1(x1)1-»S2(x2)1-"[S1(xl)-e

+S2(x2)-e

- l]-'/*.

In case of n variables, a straightforward generalization is:let Ui, i = 1,..., n be a family of gamma variables with mean 1 — p and the same variance (1 — p)9, and W a gamma variable with mean p and variance p6. Then, Wi = Ui + W ~ T(g, i ) , i = 1 , . . . , n and : S{xu...,xn)=W!=lSi{xi)l-p[^=lSi{xi)-f>

-n + l}-P'e .

Yashin and Iachine [230] give a more general formula where the variances of the Ui, i = 1 , . . . , n can be different. K is the UFB, G is an independent bivariate distribution with exponential marginals: Explicitly K(u,v)

= min(u,v)

0 < u,v < 1

and fmin(uw\vWa).ivie~v'1.W2e-w'.dw1dw2.

F(u,u) = f

In this case the marginals are: /•OO

Fi(u)

=

/

u^.toie-^.diui

=

/

/»oo

w 1 e- u , l ( 1 - '° 9 u ) dwi

Construction

=

83

of a Copula

^ — 1 — logu

(4-26)

andF 2 («) = i r ^ . After a number of transformations we obtain :

This expression involves no parameter, but its convex combination with the independence case gives :

F'(u,v)

=

(l-a)F1(u)F2(v)+aF(u,v)

= «<-«(•'(i^») "<-<'• (4) K is the lower Prechet bound, G is an independent bivariate distribution: Explicitly K(u, v) = max(u + v — 1,0), 0 < u, v < 1 . Marshall and Olkin [152] provide an example where G possesses independent exponential marginals. They arrive at the distribution: F(u,v) = Fifa)

+ F2(u2) - 1 + B(—l—, —ip-) £>i{ui) b2(u2)

where B(.,.) is the beta function. This method of using mixtures of distributions has been developed by Joe and Hu [ill] for the case of max-infinitely divisible distributions (i.e distributions for which all the positive powers of it are also proper distributions functions). A particular case of these distribution are the extreme value distributions for maxima. Joe and Hu [ill] generate many multivariate (not only bivariate) families of distributions functions with positive dependence and study properties of closure properties in these family as well as concepts of dependence including the tail dependence.

84

Copulas

4.6

Archimedean Copulas

4.6.1

Definition

and basic

properties

An important particular case of copulas are Archimedean copulas (we note again that frailty models are their particular cases). Genest, Nelsen and Sungur among others, studied these copulas in great detail [81], [83], [160], [212]. If ip, called a generator, is a convex decreasing function, with a positive second derivative, and ip(l) = 0 and such that
~\x) = { 0

for 0 < x < (/?(0) for ip(0) < x < + 0 0 .

An Archimedean copula is then defined as: C(u,v) =
(4.27)

Remark 1: The existence of the second derivative p'(C)D2C

=
(4.28) (4.29)

Therefore D\{C) = ,(Q\, and differentiating with respect to the second variable we obtain an explicit expression for D\D2C : DID2C

=

ywv)D*c 2
y»y"(CV(t;)
(4.30)

Remark 2: The convexity and the decreasing property of

Archimedean

85

Copulas

to verify using Eq. (4.30), and the fact that the convexity is equivalent to an increasing first derivative, i.e. positivity of the second derivative. Remark 3: With the same generator tp, one can define a complementary copula by using the survival functions S, Si and S2 in place of C, U and V. Indeed starting from Eq. (4.6): C'(u,v)=u

+ v-l

+ C(l

-u,l-v)

the survival function of C" is S(u, v) = C'{u, v) = C(l - u, 1 - v). Hence one can define S(u, v) = (p'1 [tp(l -u)+

cp(l - v)] .

i.e. S(u,v) = ip-^S^u))

+
Remark 4: The generator

:
Genest [81] has shown that the probability of this set is — (p(0) /
/ /
-fy(,*lW)--m]

= 1+

w>

^ j ^

(431)

where c(u,v) is the density of the copula, and using Eq. (4.30), with the change of variables z = C(u, v) and t = u we verify the assertion. 4.6.2

Examples

(1) Let the generator be ip(t) = (1 - t)a a > 1

Copulas

In this case : C(u, v) = max(0,1 - [(1 - u)a + (1 -

v)a]1/a).

The lower Frechet bound is attained when a tends to one, and the upper when a tends to infinity. (2) The Clayton family [46] 1. This family (also studied by Cook and Johnson [48]) is constructed from a frailty variable W having a Gamma distribution with the Laplace transform 1. In this case : Ca{u,v) = (u1~a+v1-a

-1)T^

a>\.

As the parameter a tends to infinity, dependence becomes maximal, and as a tends to one, the pair (U, V) becomes independent. It is possible to extend this family to have negative dependence, using the generator tp(t) = 1 - tl~a 0 < a < 1. In that case C(u,v) = max{v}~a + vl~a - l , 0 ) 1 / 1 _ a . As a -> 0 the distribution tends to the lower Frechet bound. Each member of this family has LRD dependence as it is shown Chapter 3, Section 3.2.8. But the family is not ordered by the DTP(0,l)-ordering, see Chapter 3, Section 3.6.6. (3) The Frank family [72],[82] In this case

V(0=ln£f^.0 (<*u - l)(av - I) C(u,v) = logQ [l + '- ^ - ^ '-

The dependence becomes maximal when a tends to zero and independence is achieved when a = 1. It is possible to extend this family to the case of negative dependence with a > 1 (see Meester and MacKay [154]). In this situation as a tends to infinity the copula tends to its lower Frechet bound. (4) The Gumbel-Hougaard family [88] [99] (also known as bivariate logistic extreme value distribution).

Archimedean

87

Copulas

^(<) = ( - l n ( i ) ) 1 / a , 0 < a < 1. This family is obtained by the Laplace transform of a positive stable distribution (<^_1(i) = exp(-ta)). It plays a prominent role among extreme value distributions. It is the only Archimedean copula which is also an extreme value copula. We have: C(u,v) = exp ( - [ ( - I n H ) 1 / " +

(-]n(v))1/a]a)

As a tends to zero, the dependence becomes maximal, and as a tends to zero the pair (U, V) becomes independent. No extension to negative dependence is available. (5) The Ali-Mikhail-Haq family [4]: Setting y a (f) = ( l - a ) - 1 I o g [ 1

+ a(

f*~

1)

]

,|a|
we retrieve the Ali-Mikhail-Haq family (already mentioned in Section 4.5.6). This family does not attain the Frechet bounds. This distribution as also the three preceding ones, which come from a Laplace transformation verify the strong property of LRDdependence (Chapter 3 Section 3.2.6). (6) The lower Frechet bound of a copula is Archimedean. Specifically C~(u,v) = max(0,u + v — 1) is generated by tp(t) = l-t,0
A characterization

of Archimedean

copulas

(1) The Abel (1826) [2] criterion The theorem below, due to Genest and MacKay [81] provides a characterization of Archimedean copulas. Its proof involves an identity related to symmetric bivariate functions proved by Abel

88

Copulas

in 1826. For this reason Genest and MacKay refer to his theorem as the Abel criterion for copulas. Theorem : A copula C is Archimedean if (and only if), there exists a mapping / : (0,1) -> (0, oo) such that : 9C{u,v)

,,

•.

KJ^=M

V(u,,);0
dv

The function

W

^

'

To show the converse, Genest utilizes Abel criterion. [2] We can now verify that FGM family is not Archimedean. Indeed in this case : ^ _ . ( 1 + 0 ( 1 - i , ) ( l - 2u) dC M u(l + 6{1 - u)(l - 2v) dv

which is not of the form 4^f, except when U and V are independent (i.e 6 = 0). (2) The Ling (1965) theorem One of the earliest results on Archimedean copulas is Ling's theorem which states that Archimedean copulas are the only copulas which satisfy the two conditions : (a) associativity : V(u,v)C(C(u,v),w) (b) V u € ] 0 , l [ , C ( u , u ) < u . 4.6.4

=

The limit of a sequence of Archimedean

C(u,C(v,w))

copulas

Let Cn ,n £ N n > I be a, family of Archimedean copulas, and let , the set of functions,

Archimedean

89

Copulas

which are continuous, strictly decreasing and convex, with a continuous second derivative. The functions (pn belong to . Proposition 1 ([81]): Assume that the copulas Cn are absolutely continuous with respect to the Lebesgue measure. The limit C is an Archimedean copula if and only if, there exists tp £ <J>, such that for all s and all t in [0,1] :

Km

~ ~ ^77Y = ^WY

(4 32)

'

It Eq. (4.32) is valid, then there exists a sequence of constant c n , such that Zimn_>00c„y>n(i) = (p(t) Proposition 2 ([81]): Let cpn, n > 1, be a sequence of functions belonging to $ , and let Cn be the sequence of associated copulas. Then: Va; Vy 6 [0,1] lim Cn(u,v) =

min(u,v)

n—>oo

if and only if limn-¥00f',,tl

= 0 for all t in [0,1].

Example: Choosing (pn(t) = t~» — 1, and cn = n, we have : lirrin^oorvpn^) = -logt .

(4.33)

Hence: Urrin^ooCniUyV)

= UV .

The sequence Cn(u,v) is a subfamily of the Clayton family mentioned in the previous Section. 4.6.5

Characterization diagonal

of Archimedean

copulas by their

copulas

Equation
Copulas

90

Conversely, given D, the diagonal of an Archimedean copula, is it possible to define uniquely a generator
£>:[0,1]->[0,1], D(0) = 0, D(l) = l; D(u) is a strictly increasing function in u ; D{u)
n->oo

^ - > ) n

D- (u)

y

( l ) - 1

However (p(l) = 0 and tp(D~n(u) = 2~n(p(u), hence :

n-)-oo D

n

Therefore if lim,!-^ 2n(D~n(u)

(u)

—1

— 1) exists, we arrive at

Vu €]0,1[
- 1) .

n-foo

We know that ip(u) is defined up to a constant, that is ap(u), where c is a positive constant, generates the same copula. Hence utilizing the fact that (p'(l) is always negative, we have the relationship :
- 1) .

(4.34)

Archimedean

91

Copulas

- We now sketch a proof of the sufficiency condition. With

+ D~n(v) - 1).

n—foo

Remark : if the condition (5) is not fulfilled , that is if Vu G (0,1),

lim 2n{D-n{u)

- 1) = oo

n—voo

one can still construct a generator ip for an Archimedean copula : Sungur and Yang [212] construct it pointwise. 4.6.5.1

Fitting an observed distribution with an Archimedean copula

Using the results of the preceding Section, and the property that a diagonal copula is the cumulative distribution function of W = max(U, V) (Section 4.1), we can fit an observed distribution to an Archimedean copula. The approach described below is due to Sungur and Yang [212]. -Given a sample (Xi,Yi), i = l,...,n, we can construct the sample Wi — max(Fin(Xi),F2n{Yi)), and estimate Gn(w), the empirical distribution function of Wi. We ought also to assume that Gn(w) is always less than w. One can, in a given Archimedean family of copulas indexed by a parameter 6, estimate

92

Copulas

this parameter, applying, for example, the method of moments. Explicitly we shall choose the family 1 (Section 4.6.2) with the generator tp{u) = (1 - u)e,

0 > 1.

In this case the derivative of the diagonal copula D(w) =
mw) = v w

=

mi-*)9-1

= 2l/.

Therefore D(w) has here a uniform distribution on a part of (0,1). The expectation of W is: E(W)=

[

2*wdw =

l-2-i-1.

-

./[l-2 ?,l]

'ywi This expectation is estimated by W = - ^ — , and one deduces the estimator of 9 to be log2 o = - log2{l - W) The difficulty with this method is that, first by we have no criteria to choose a priori one family or another (although we could compare a posteriori the empirical distribution Dn(w) with the fitted distribution Dg(u)). Second the expression of D'(w) as a function of the parameter may not be so simple as in the example and consequently the expectation of W can be intractable. The method described in the next Section, due to Genest and Rivest [83], allows us to fit the observed distribution with any Archimedean copula in a simple manner. The comparison between the empirical and the fitted distributions is carried out by means of a x 2 test. In the last chapter we shall provide criteria for choosing the family. 4.6.6

Characterization of an Archimedean copula by the cumulative distribution function of Z = C(U, V)

The key results (due to Genest and Rivest[83]) in this section are: (I) The function K(z) — z — irk is the cumulative distribution function of the variable Z = C{U, V). Therefore knowing K(z), one can

Archimedean

Copulas

93

in principle retrieve the function (p(z) and hence the Archimedean copula. (2) The function K(z) can be estimated by means of the empirical distribution function Kn(zi), where Z{ is the proportion of the pairs (Xj,Yj), which are less than or equal to the pair (Xi,Yi). (3) The empirical function Kn(z) can be fitted by the distribution functions K^(z) of any family of Archimedean copulas, where the parameter 6 is estimated in such a manner that the fitted distribution has a coefficient of concordance (r) equal to the corresponding empirical coefficient (T„). More precisely, Genest and Rivest [83] prove the following proposition: Proposition: Let (U, V) be a random vector from an Archimedean copula, with generator tp. Set W = ^tm+l/v)
an<

^ % = C(U, V). Then

(1) W is distributed uniformly on (0,1); (2) Z is distributed as K(z) = z — X(z), where X(z) (3) Z and W are independent.


These results can be used in other contexts as well, for example for generating a sample (U, V) from a given Archimedean distribution. Proof of the proposition: Assume that C is absolutely continuous, and let g(w,z) be the density of (U,V) and G(z,w) = P(Z < z,W < w), then G{z,w) = / / g(w,z)dwdz Jo Jo

= / / c(u,v)\ J J

' \dudv , d{z,w)

where c(u, v) = -^^ is the density of the copula and J"'^x is the Jacobian of the transformation (u,v) -> (z,w). Here we have f}f\ • Proceeding as in Section 4.6.1 (Eq. 4.31), we have : d2z dudv

-v'(u)
Hence:

=

Jo Jo wK(z).


tp'iuWiv)'

The last relationship proves the proposition.

94

Copulas

4.6.7

Archimedean

copulas with two

parameters

It is easy to prove that by means of a generator tp of an Archimedean copula, it is possible to construct other families of generators [161]. Indeed if * is the family of all the Archimedian copulas then: (1) If ip e # , then 1. (2) If if 6 * , then <pa is such that tpa(t) = (y{ta)) € * , V 0 < a < 1. (3) Using these two properties successively, we obtain that <pap(t) — 0 generates an Archimedian family (Clayton family). (2) For ip(t) = -ln(t) and /? > 1, we obtain ipp(t) = (-ln(t))fi, which corresponds to Gumbel-Hougaard family. (3) The generator ipap{t) = (ta - 1)^ with a > 0 and /3 > 1, allows us to construct the Archimedean copula:

Ca,0(u,v) = {[(u~a - \f + (y~a - \f$ 4.7

+ l}-i .

Archimax Copulas

Recently Caperaa et al. [38] have defined a new family of copulas for which Archimedean copulas and extreme value copulas are particular cases. 4.7.1

Extreme

value distribution

and extreme

value

copula

Let (UI,VL),... ,{Un,Vn) be a random sample from an arbitrary copula C and let Umax = max(U\,...,Un) and Vmax = max(Vi,...,Vn) be the two maxima in the sample. The extreme value distribution is the limiting distribution, if it exists, when n tends to infinity of (Umax, Vmax)- However: P{Umax

Cn(u,v)

< U,Vmax
and the marginals of this distribution are: P{Umax
.

95

Archimax Copulas

Hence the copula associated with the extreme value distribution of C is: = limn-u>0Cn(u",V")

Cmax{u,v)

.

In that case we say that C belongs to the domain of attraction of Cmax • 4.7.2

Definition

of Archimax

copulas

Following the work of Pickands [168], Caperaa et al. [37] use the general form of a bivariate extreme value copula : CA(U, V)

= exp

(4.35)

\\og(uv)

where A is a convex function [0,1] •-»• [1/2,1] such that max(t, 1 — t) < A{t) < 1 for all* € [0,1]. The family of Archimax copula is then defined as: CV,A(U,V)

=tp

{
*{U)

Mu) + f(v).

where A is the convex function defined above and ip is the generator of an Archimedean copula. If A = 1, we retrieve the Archimedean copulas, and if (p(i) = —ln(t), we retrieve the extreme value copula with the Caperaa formulation (Eq. 4.35). Remark: This procedure to generate a bivariate copula is a particular case of Marshall and Olkin's generalization (Section 4.5.6, Eq. 4.22), where the function K is here the extreme value copula CA* , and the mixture distribution G has for its Laplace transform the generator (p~l. 4.7.3

Construction predetermined

of bivariate distributions domain of attraction

belonging to a

The aim of Caperaa et al. is then to construct families of Archimax copulas belonging to the domain of attraction of a predetermined extreme value copula CA' • For this an additional condition is needed on the generator tp : 1. That is : limt^ac

; ~ ? v = s~m, Vs > 0.

One then writes
96

Copulas

Proposition (Caperaa et al.): The Archimax copula CV
A*{t) = (tm + (1 - t)m)1/mA^m

(

*™—A

This attractor may be regarded as an Archimax distribution with generators A* and ip(t) = -ln(t) ( or with generators A and tp*(t) {-ln(t))m). Furthermore, A and A* coincide iff m = 1. Conversely, given
A(t) = (t1/m + {l-t)1/m)m.{A*{— w v *• V t/m

ti/m

+

However condition on m is imposed to assure that domain of attraction of C V . 4.7.4

\

—-)] {l-t)1/m'J CVIA

m

.

belongs to the

Examples

(1) The Gumbel-Hougaard family It is the only family which is both Archimedean and extreme value copula. (2) Ae{t) = e.t2 - 0.t + 1, 0 < 0 < 1 and tp(t) = tl~a - 1 a > 1. Here Ag (t) is known as the generator of the Tawn mixed model and
4.8

Copulas with Discontinuity Constraints

Wei et al [223] have studied copulas whose probability measure is concentrated on a set of measure zero, called "piecewise additive copulas" and copulas whose density functions are piecewise constant (that are locally independent) which they call "piecewise quadratic copulas" and a particular case of the preceding copulas for which the density function is zero on some "holes" of the square unit and piecewise constant elsewhere.

Copulas with Discontinuity

4.8.1

Piecewise

additive

97

Constraints

copulas

Suppose that the unit square is partitioned into measurable closed sets: [0, l ] 2 = Ui^i Ai whose boundaries DAi are piecewise differentiable curves with finite length and Ai f] Aj C DAi fl -^A? for all i ^ j . Theorem: The joint distribution of (U, V) is singular iff the copula is piecewise additive, i.e. on each partition set Ai, we have C ( u , i O k = C i ( u ) + C2(i>), where Ci(u) and C2(v) are some increasing functions. Proof (1) Necessity If the joint distribution is singular, then there exists a partition on the unit square such that, the probability measure is concentrated along the boundaries of the partition set. Thus within each partition set the increment of the cumulative distribution function is the sum of increments of conditional marginal distributions :

C(u + Au,v + Av)\Ai

=

[C(u

+Au,v)-C(u,v)}

+

[C(u,v +

=

C1(Au) + C2{Av).

Av)-C(u,v)}

(2) Sufficiency Since an additive copula possesses no density and since the cumulative distribution function increases by increments, the probability measure is concentrated on the boundaries of the partition set. Example: Let (X, Y) be uniformly distributed in L\ norm on the unit sphere {(x,y) : \x\ + \y\ = 1}. The corresponding copula is then concentrated along the lines y = x±\ and y — —x + l±\. Therefore there exists a partition of the square unit into five regions : 2

C(u,v) = < 0 u v u + v-1

4

u

I

u

21 ^ 1

2I — 2

if |u - | | + |u - | | > | 0
98

Copulas V 1

Fig. 4.1

(l-X)a

H2-X)a

X.a

(1-X)a

A piecewise quadratic copula.

Thus C(u,v) is piecewise linear. Remark : The uniform distribution on the circle {x2 + y2 = 1} has the same copula ([223]).

4.8.2

Piecewise

quadratic

copulas

A piecewise quadratic copula has a piecewise constant density. For example: let 0 < A < 1 and 0 < a < 537, and a partition of the unit square into four rectangular regions as given in Figure 4.1.

4.8.3

Quadratic

copulas with

holes

Definition: Let (U, V) be a continuous random vector over the unit square J = [0, l ] 2 . Let 5 be the support of the density function. Let us call DM, the boundary of a set M. A connected set yl is a hole if A C J — S and

DAC(DJ\JDS). A simple example: consider the family of copulas with a single hole A and the density constant over U - A. If the hole has width a and height /? then the density over J - A is given by 3-^3 (Fig. (4.2)).

Copulas with Discontinuity

Constraints

V

T3

Dl

Tl

Bl

B2

T3

D2

T4

a

-* Fig. 4.2

4.8.3.1

^

A rectangular hole within a copula.

Admissible rectangles

Wei et al. propose a heuristic method (inspired by a conterexample of Stoyanov [211]) for constructing a copula with holes. Since the marginals of a copula are uniforms, the idea is to shift the omitted mass of holes along one axis, next along the other axis and again along the first one as schematically indicated in Fig. (4.3). Before describing the method we have to define the concept of an admissible rectangle, i.e. a rectangle possessing the holes. First we shall consider a single rectangular hole of with width a and height /?.

From the definition of a hole, the probability measures of the rectangles Di, D2, Bi, and B2 must be positive. We shall denote them by di, d2, bi, b2. Since the marginals are uniform, we have: 1 x a = di + d2 1 x /3 = bi + b2. However the probability measures of the rectangles T2, Ti, T3 and T4 must

100

Copulas

Fig. 4.3

Holes inside an admissible rectangle

also be strictly positive, so that: l-a-8>0 From these constraints, we deduce that: a.B <max{a{l

-a),8(1

-/?)) < - .

A rectangle which satisfies these constraints is called admissible. If a and 8 satisfy these constraints, we can construct a copula with a probability mass equal to a on the two rectangles (Di, D2), a mass equal to B on the two rectangles (Bi, B2) and the remainder of the mass 1 — a — B situated on Ti, T 2 , T3 and T4. 4.8.3.2

The squeeze algorithm

We shall consider now more generally a construction of copulas with holes that are contained within an admissible rectangle.

The construction algorithm is as follows (see Fig. 4.3):

Copulas with More than Two Variables

101

(1) Start from the uniform distribution on the unit square. Suppose that the mass measure of holes is s. Set the density to zero over the holes. (2) Distribute this mass over Di |J D2 such that the marginal density along u will remain 1 for u 6 [0,1]. (3) Remove the mass s from the Tj , i = 1,..., 4 so the marginal density along v will remain 1, for v £ [0, c] (J[c, 1]. (4) Distribute the mass s over B\ \JB2, so the marginal density along u for u £ [0, a] X [b, 1] is 1 and along v for v G [c, d\ is also 1.

4.9

Copulas with More than Two Variables

It is quite tempting to use the machinery of copulas to construct multivariate distributions with given marginals for m (m > 2) variables. There are many ways to generalize with respect to the marginals that we have decide to fix: fixing m univariate marginals and searching for a mapping [0, l ] m M- [0,1], which defines a dependence structure, or fixing two marginalsFp(xi,X2,•••,zP) and Gq(yi,y2,...,yq) withp+q — m and searching for a mapping [0, l] 2 >->• [0,1], or fixing some 2-dimensional marginals, etc. However there are compatibility constraints on the marginals. For example in the class of 3-dimensional copulas, the marginals C12, C13 and C23 overlap and are therefore not completely independent from one another. Furthermore a given family of m-copulas (m > 2) does not attain its lower Frechet bounds. The tool of copulas is less universal in the case of m (m > 3) variables than in the case of two. Specifically Genest et al. [84] have shown that given C a 2-copulaand two marginals Fm(xi,X2, ...,xm), and Gn(yi,y2, ••yn), the only copula which satisfies that H(xi, ...,xm,yi, ...,yn) — C(Fm(x),Gn(y)) is a cumulative distribution function with the marginals F and G for any F and G, is the independent copula. To show this result Genest et al. utilize an example where F, of dimension 2, is precisely the Frechet lower bound max{0, Xi+x2 — 1}- However if we impose some restrictions on the marginals we may hopefully be able to construct families of multivariate distributions.

102

4.9.1

Copulas

m-dimensional

Archimedean

copulas

Jouini and Clemen [120] propose to iterate the bivariate Archimedean copulas to obtain any m-dimensional Archimedean copula using the property of associativity (Ling's theorem, see Section 4.6.3). The procedure is as follows. Let C be a given two-dimensional copula. Define C2 = C and recursively Cm — C ( C m - i , u m ) , m > 2 This procedure works well in the case of frailty models, as we have already seen in Section 4.5.6, (Eq. 4.24). If ip"1 is a Laplace transform, the expression Cm becomes: C m (ui,..., um) =
(4.36)

Moreover, as it was shown by Schweizer and Sklar [196], in order to extend to all n-dimensional distributions on the positive region,
= [u\-a + ... +u1ma-m

+ l]^,

a> 1.

Now for m = 2, the lower Frechet bound C(u, v) — max(0, u + v — 1) is Archimedean (with the generator
— max(ui + u-i + u3 - 2,0)

which is not a cumulative distribution function (see, e.g., Problem 748 (1976) American Mathematical Monthly, 83 and Solution: American Mathematical Monthly, 85, (1978) p. 393). 4.9.1.1

An application

Meester and MacKay [154] have proposed to use Frank copulas model for fitting models to clustered binary data. The size of the cluster is n (in general n > 2). This is possible since Frank model is obtained by Laplace

Copulas with More than Two

Variables

103

transform (Eq. (4.36)). The association parameter a of the Frank copula model provides a measure of the strength of the within-cluster association. The advantage of Frank model is that even with binary marginals, the parameter a is almost independent of the marginal probabilities, which are also modeled. Tests on the parameters are performed using maximum likelihood method and are computationally straightforward. Tregoiiet et al. have applied this idea to modelling the joint distribution of a binary trait (a disease status) within families. They decompose a family into two sets (parents and offspring), each of them characterized by an association parameter (aFM and ass respectively). The marginals probabilities are modeled by means of a logistic representation. They perform a segregation-linkage analysis of levels of plasma angiotensin converting enzyme (ACE) dichotomized into two classes. The key steps are the following: (1) Consider a family in which a binary trait (disease status) is measured. Let Y = (YF,YM,YI, ...,1^) the vector of the trait for the father (F), the mother (M), and the n children. Similarly x = (xp, XM, XI..., xn) be a vector of covariates of the family. (2) Suppose that, conditionally on an individual's own covariates, an individual's status is independent of the covariates of the other family members. The joint probability of the trait given the covariates can then be decomposed into two probabilities : P{Y/x)=P(YF,YM/xF,xM)-P(Y1,...,Yn/YF,YM,x1..xn) (3) The two probabilities are modeled by two different Frank's models (with parameters aFM and ass respectively) since (YF,YM) and (Fi,..., Yn) are considered each to be equicorrelated data. For example for the children we have: P(Yi
= •••> Fn(yn))

loga [l + (a - l)n?=1(

= a

_

i

)j .

(4) The marginal distribution functions are modeled by a logistic representation.

104

Copulas

For example, for a mother we have 0

FF(YF/xF)

={

if YF = - 1

1+e^.mr

ify F = 0

1

if YF = 1

where A is the baseline hazard and j3 is the vector of marginal regression parameters of covariates. For a sibling:

FF(Yi/xi,yF,yM)

[ 0 1 = < i+^+^..i+TJ. 0.»F+->M0.BM

I

1

if Yi = - 1 if 1* = 0

if Yi = 1

where 7 F O and 7MO are the regression coefficients, for the parents phenotypes, that characterize the familial aggregation between the parents and the offspring. (5) The joint probability mass function, for example for the pair (YF,YM), is then formally written as: P(YF,YM/xF,xM)

=

CaFM

{FF(YF/XF),FM{YM/XM))

-CaFM {FF{YF/xF),FM(YM l/xM)) ~CaFM (FF{YF l/xF),FM(YM/xM)) +CaFM (FF(YF - 1/XF),FM(YM l/xM)). (4.37) Formulas for the children are similar. The association parameter ass m the model for the sibs characterizes the residual aggregation between sibs after controlling for the parents-offspring effects (which is represented by the parameters jFo and 7MO)- It can be due, for example, to the shared environmental factors. Actually, the model is more complicated and incorporates a major-gene effects. Therefore the two association parameters aFM and ass correspond to a residual effect between the spouses, and between the sibs after controlling the major gene effects. Conclusions of this analysis are similar to those reported in an earlier analysis where the data of the ACE levels were not dichotomized, however there is a loss of power due to the dichotomization and the residual effects of association could not be shown to be significant.

Copulas with More than Two

4.9.2 4.9.2.1

Generation marginals

of a ^-dimensional

105

Variables

copula from its

2-dimensional

Compatibility of marginals

Suppose that we are searching for a copula C123, with fixed marginals C\%, C\z, C23. We have to verify (using a shorthand notation) that : Cl2 = /

Ci/3C2/3dll3

where C1/3 and C2/3 are the conditional distributions given u3 and where the integration is taken on the domain of definition of U3. The two other relations obtained by permuting the subscripts {1,2,3} ought to be verified as well. 4.9.2.2

Truncation invariance

Chakak and Koehler [40], and subsequently Sungur [213] defined a class of three-dimensional truncation invariant copulas which are determined solely by their two-dimensional marginals. Using this procedure, it is therefore possible to construct a single 3-dimensional copula, choosing 2-dimensional marginals provided that the marginals are compatible. A 3-dimensional copula is truncation invariant in one variable, if the dependence structure between the two other variables is not affected by the truncation of this variable: for all a3 G [0,1] the distribution function of {Ui,U2)/U3 < a3 is the same as the marginal distribution function of (Ui, U2)- In that case the 3-dimensional copula satisfies: Ci23(ui,U2,U3)

= C12

Ci3(ui,U3)

C 2 3 (U2,U 3 )

u3

u3

u3 .

(4.38)

Namely, the joint distribution of {U{r, Ul2r) = [{U1,U2)/U3 < a3] is:

F(Uf,Ut) = m < ^ Y 3 < a 3 ) " F{U3 5 a3) The marginal distributions of U\T = Ui/U3 < a3, i = 1,2, are: Ptr, bi (Ui)

x _ PJUi < Uj,U3 < a3) _ ~ P(U3 < a3) ~

Ci3(ui,a3) a3 •

< 4 - 39 )

106

Copulas

If Ca3 denotes the copula associated with {U{r, U2r) namely: = CM{Ulr,F2W)))

F(U?,U?)

= Ca3 f C l 3 K ' Q 3 ) , \ 03

C23(

"2'a3)) , a3 J

then : Ci23{ui,u2,a3)

=

P{Ui < Ui,U2 < u2,U3 < 03)

= F(UiT,U?)a3 _ n fCi3(ui,a3) C23(u2:a3)\ — °a 3 I , a3 • \ a3 a3 / If this result holds for all a3, then Equation (4.38) is valid. If this property is also valid when conditioning on the two other variables, the copula will be truncation invariant. Examples: (1) pairwise independent bivariate copula : Cij = Ui.Uj i ^ j G 1,2,3. In this case n

,

Cl23(Ui,

U2,U3)

\

n

,Cij{ui,Uj)J

= Cjk{-^

Ui

—,

Cik(ui,Uk), Ui

).Ui =

UiU2U3.

Hence for this construction pairwise independence implies mutual independence. (2) Plackett's family (Chakak and Koehler [40]): In the Plackett's copula [169], the odds-ratio : C12(u,v)C{u,v) {u - Ci2(u,v))(v - Cx2(u,v)) is constant. The associated copula (cf Section 4.4 Eq. (4.13)) is :

C12(u,v) =

G(u,v) - ,JG2(U,V)

2 (a 1 2

- 4a 12 (o:i2 - l)uv

- 1)

ai2 >

°

with G(u,v) = 1 + (ai2 — l)(u + v). Construction of the trivariate copula by conditioning on u3 gives : Ci23(u,v)

=

C12

Ci3(ui,u3)

C23(u2,u3) "3

u3

u3

107

Copulas with More than Two Variables _


~ 4ai2(«12 ~ l)C , i 3 C l 23 2(«i2 - 1) (4.40)

where G12.3 = u3 + ( a i 2 - l)(Ci3 + C23). One can also obtain this copula by stipulating t h a t t h e conditional odds-ratio given u3 is constant for all ui, U2 .This condition yields _ Ci23(w3 — C13 — C23 + C123) (C13 — Ci23)(C,23 — C123) Solving this equation, by stipulating t h a t 0:12/3 * s constant, we then retrieve Eq. (4.40). However, for t h e Plackett's copula, conditioning on other variables t h a n u3 does not result in t h e same copula. (3) Clayton's family : This family satisfies truncation invariance. Namely C{uuu2,u3)

= (u\-a

+ u\-a

+ul~a

- 2 ) ^

(4.41)

can be rewritten as: C(wi,u2,u3) = I r ("i- a +"r o , - i > i / i "°', _ri \[—*—^ ]°

+

Q r"2" +"3 _ °- i ) i / i _ °', 7 ~11 i- 2 —"-^i 1"-1 /

U3

'

which is n

,C\i{ui,Uz) u3

C23{u2,U3) u3

This equality remains valid if we permute the indices {1,2,3}. (4) Three parameters Clayton's family: It is possible to generalize the preceding formula with on, i = 1,2,3, being associated with the bivariate copulas C\2,C\3 and C23. If C y ( u i > U i ) = (u\-a< +u)-a'-1)^

, at > 1 .

Provided that the marginals Cy, i ^ j are compatible, one can construct the copula :

C{ui,U2,U3)

=

108

Copulas

(u\-ai + Ul~ai - !)£!!• + (UJ-«* + „l-°2 - l ) ^ f _ u l-«3

-1

(4.42)

Note that Eq. (4.42) was obtained by conditioning on uj. Analogous expressions could be obtained by conditioning on u2 and u3. (5) Lower Frechet bound copula : For C~(iii,Uj) = max(tij +Uj — 1,0), we can construct

C{Ui,U2,U3)

= C U3

=

U3

max[max(ui + u3 - 1,0) + max(u 2 + u3 - 1,0) - U3,0].

(4.43)

This procedure can of course be generalized to obtain m-dimensional copulas (m > 3) using only the two-dimensional marginals, and the property of truncation invariance. We note that since the formal three-dimensional lower Frechet bound is not attained Eq. (4.43) may be viewed as an alternative version.

4.9.3

Linkages

In the case of two variables, a copula allows us to separate the effects of the univariate marginals from the effect of dependence. Using the same idea and being inspired by the setwise dependence, Li et al. [143] have proposed extending this approach to the case when (Xi,X2, ...,Xfc) are k random vectors of dimensions mi,m,2,---,'mk respectively with the marginals Fi,F2,...,Fk. For each random vector X^ = (Xn,Xi2,...,Xiki), they associate a vector U; = {Un,Ui2,..-,Uik.) where the U^ are independent and uniform on [0,1]. The linkage is then defined as the joint distribution function L of the vector (Ui,...,Uk)- Thus the linkage is useful in studying the dependence properties between the X; separate from the dependence properties within each X;. However decomposition of each vector Xj into Uj is not unique, it depends on the order of the subscript ii,—,iki, a n d therefore the linkage L is also not unique. How does this construction work? Let X = (Xi,..., Xk) be a k-dimensional vector with an absolutely continuous joint distribution function F and the marginals Fi ,i = l,...,k. The procedure is as follows : (1) U1 = F1(X1)

Copulas with More than Two Variables

109

(2)

U2=F2/1{X2/X1=x1) where F2/\ represents the conditional distribution function of X2 given Xi = x\. (3) Ui = Fi/h2ii^1(Xi/x1,...,xi-i) In this construction the variables Ui are independent and uniformly distributed. Example : Let X = ((Wi, W2), (Zi,Z2)) be a four-dimensional normal random vector with mean 0 and the correlation matrix given by : /

1 pw Pw 1 £ = P P V P P

p p \ P P 1 Pz Pz 1 /

The constraint on the parameters is : — 1 < p < | - \ / ( l + pw)(l + pz) in order to assure that S be a semidefinite positive matrix. We have : W\ ~ N(0,1) ; W2/(Wi = w\) ~ N(Pw.Wl, 1 - p2J, Zi ~ N(0,1) , Z2/(Zl = Zl) ~ N(pz.zi, 1 - p2z). Thus the joint distribution function L is : ( U, \ U2 U3

(

$(Wi)

*(Z1)

\ where $(.) is the standard normal distribution function. To arrive at a more indicative form of L, Li et al. [143] present the distribution function of the vector ( S - ^ f / i ) , * - 1 ^ ) , * - 1 ^ ) . * - 1 ^ ) ) which is a normal distribution function with mean zero and the correlation matrix: £ = where :

/ S2

S2 I

Copulas

110

4.10 4.10.1

Simulation Procedures The general

case

To generate a sample (Ui,Vj), i = l,...,n from a copula C(u,v), one uses the fact that Cu(v) = C(V/U = u) is a distribution function, and that Z = CU(V) obeys a uniform distribution on [0,1]. Since U has a uniform distribution, its density on [0,1] is 1 and thus Cu(v) = g^ • The procedure is as follows : (1) First step : Generate two random variables U and Z, independent and uniform on [0,1]. (2) Second step : Calculate V = C~1(z). The pair (U, V) has the desired distribution. This procedure works well but it necessitates having an analytical expression for C " 1 . 4.10.2

Archimedean

copulas

In the case of Archimedian copulas, one can adapt the procedure described above. This method is due to Genest and Mackay [81]. Specifically: (p(C) =
An auxiliary variable W = C(U, V) is calculated by :

H^r1 (^) M^r1 (^)

(^

where (<^') _1 1S the inverse function of the derivative of (p. The procedure is therefore as follows: (1) First step: Generate two uniform and independent random variables U and Z on [0,1]. (2) Second step : Calculate W, using formula (4.44). (3) Third step :

Simulation

Calculate V = y-l[ip{W)

111

Procedures

- y{u)}.

This procedure works well for the Clayton and Prank families. However for the Gumbel-Hougaard family we don't have an analytical expression for For Gumbel-Hougaard family one can use the procedure developed by Lee [141]. Here one uses the fact that for any Archimedean copula, T = (in+ \v) n a s a uniform distribution independent of Z = C(U, V) (Section 4.6.6), and that for Gumbel-Hougaard family, Z\ = ((p(Z))a is distributed as a mixture of two gamma variates with the density c(z) = (1 — a + a.z)exp(-z); here a the parameter of the copula. One then retrieves U and V from T and Z\. 4.10.3

Archimax

distributions

In this case the algorithm uses the same auxiliary variables Z = C^^iU, V) and T = /J), \v\ as for Gumbel-Hougaard family described above. Specifically, the joint distribution function of (Z, T) is

P(Z < z,T< t) = Kv(z) (t + t(l-t)^±

t

+ \v(z)£

-^^dA'(t)^j

where A' is the derivative of A, K(p(z) = z — Ork, is the distribution function of the variable Z and ip' stands for the right derivative wherever ip is not differentiable. In particular

P(T
A'ft) t(l-t)-^.

+

Suppose now that A possesses the second derivative A" continuous everywhere on (0,1), and let h(t) be the density of H(t). Define

..

t(l-t)A"(t)

n

where it can be shown that 0 < p(t) < 1 [85]. The conditional distribution of Z given T = t is then: P(Z < z/T = t)= p(t)z + (1 - p(t))Kv(z)

,

This conditional distribution of Z given t is therefore a mixture of the univariate distribution Kv and a uniform distribution on (0,1).

112

Copulas

Thus the procedure is as follows : (1) generate T from distribution H; (2) given t the value of T, draw W from a uniform distribution on [0,1]; (3) if the value w taken by W verifies w < p(t), select Z from the uniform distribution on [0,1], otherwise generate Z from distribution K

Marshall

and Olkin's mixture

of

distributions

If the form of the functions K and G are simple (for example independent variables, upper Frechet bound, etc.) one can use the following method: (1) Generate W = {Wu..., Wn) from G. (2) Generate V = {Vu ..., Vn) from K. (3) For i = l,...,n, calculate Ut = H~01(vi'Wi), where Hi0(u) = exp(-^j(u)), i^" 1 ^) being the Laplace transform of the marginal Gi, i = 1,2 of G. Details are provided in [152]. 4.10.5

Three-dimensional ance

copulas

with truncation

In this case one uses the particular structure of Ci23(u,v,w) L

C123(u, v, w) = Cxi

,

\

w

invari:

w. w

J

(1) Generate a uniform variable W on [0,1] (2) Generate the pair (X,Y) from the Cn distribution (3) From Zx = WX and Z2 = WY, determine U = C1"31/W=u,(2;1) and V

=

C

23/W=w(Z^-

An interested reader is referred to [40] for additional details.

Chapter 5

Farlie-Gumbel-Morgenstern Models of Dependence

5.1

Introduction

In this chapter, we shall summarize in some detail statistical and probabilistic properties of a popular well-known family (which seems to be quite natural -at least empirically) of bivariate dependent variables and its numerous generalizations scattered in the literature. Dependence properties of this family are closely associated with the correlation coefficient although a priori the pivotal parameter of the family is not obviously associated with this concept. Some of the results presented in this chapter appear for the first time in monographic (or periodic) form. In the author's opinion the family and its generalizations are ideally suited for various models with small or moderate dependence and do not depend on a particular physical model which may or may not be appropriate in a given situation. It should be emphasized that all the distributions discussed herein, can be viewed as particular case of the multivariate distributions obtained using the construction method developed by Ruschendorf in his pioneering paper 1985 discussed in some details in Chapter 4. In fact, it was Ruschendorf who inspired -albeit indirectly- investigations of many a distribution developed by the authors of this monograph. Order statistics from Farlie-Gumbel-Morgenstern (FGM) distributions seems to be especially attractive. Some novel results on this topic are presented in this chapter.

113

114

5.2

Farlie-Gumbel-Morgenstern

Models of Dependence

Initial Definition

Morgenstern (1956), Farlie (1960) and Gumbel (1958) have discussed families of bivariate distributions of the form •Fx(x) = = F12(xux2)

FXl,x2(xi,x2)

= F1{xi)F2{x2)[l

+ aS1(x1)S2{x2)}

(|a| < 1)

(5.1)

where i*x(x) is the joint cumulative distribution function of Xi and X2, Fjfrj) = Fxiixj) a n d SJ(XJ) = 1 - FJ(XJ); 3 = 1,2. Equation (5.1) is consistent in the sense Fx1,x2(xii00) = Fi(xi), etc. If the densities /x(-) (corresponding to i"x(-)) exist then (5.1) implies : / x ( x ) = h2(xux2)

= / i ( x i ) / 2 ( z 2 ) [1 + a { l - 2Fi(a;i)}{l -

For the densities fx{x),fi(xi) Generally, however, 1 max{(l - m i ) ( l -m2),MiM2}

and f2{x2)

2F2(x2)}}. (5.2) to exist we must have \a\ < 1.

1 < a < ~ ~ max{(l - m i ) ( l - m 2 ) , M i M 2 } '

where rrij = inf Hj; Mj = sup Hj; H) = {Fj (x) : - 0 0 < x < 00} f) {0,1}, j = 1,2. We note that if (Xi,X2) have a joint FGM distribution and Y\ = hi(Xi), Y2 — h2(X2) are monotonic increasing functions of Xi,X2, respectively, then Yi and Y2 also possess a joint FGM distribution. This is easily seen by noting that

p\

n(^<Msj))>=«*i,s2)

and P{Yj
= Fj(xj).

If Xi and X2 are each continuous, we can find transformations Yi = hi (X\), Y2 = h2(X2) so that Yi and Y2 each have a standard uniform distribution. The resulting special FGM- a copula- distribution (with the uniform marginals) was discussed in 1936 by H. Eyraud, which is, we believe, the

Regression and

Correlation

115

earliest reference to FGM distributions. (See also Kimeldorf and Sampson (1975) [127].) In terms of survival functions

Si2(xi,x2) = PI f] (x, > *i) | ; Sjixj) = P{XJ >

Xj},

(5.1) is equivalent to Si2(xux2)

= S1(x1)S2(x2)

{1 + aF1(x1)F2(x2)}

.

(5.3)

Note the symmetry with the definition (5.1). Further investigations are due to Johnson and Kotz (1975, 1977) and Cambanis (1977).

5.3

Regression and Correlation

It follows from straightforward calculations that E{X2

| X : } = E{X2}

+ aJ2 {2F1(X1) - 1},

where J2=

j

F2(x) (1-F2(x)) ,) •dx

— oo

and oo

Ji= J

FtWil-FiWdx

-OO

lim

E{X2

\Xl}

= E{X2}-aJ2,

Xi—•—oo

lim E{X2

\ X{\ = E{X2}

+ aJ2,

Xi—>oo

cov(Xi,X2)

=

aJiJ2,

and var{X2 | Xi) = var{X2) +2a{2F1(X1) -a2Ji{2F1(Xr)-l}

- 1}{K2 -

J2E[X2}}

116

Farlie-Gumbel-Morgenstern

Models of Dependence

with oo

K2 = I

xF2(x)[l-F2{x)]dx,

—oo

lim

var{X2

\ Xx) = var {X2} - 2a {K2 - J2E [X2]} -

a2jL

\ Xx) = var {X2} + 2a {K2 - J2E [X2]} -

a2Jl

Xi —>—oo

lim var(X2 Xi—>oo

If var(Xj)

= 1 and E{Xj}

= 0 (standardized variables) then

corr(Xi,X2)

-

aJ\J2.

If also X2 has a symmetric distribution, then K2 = 0 and var(X2 | Xi) = var(X2) - a2J\ {2F^{XX) - l } 2 . The cases a = 1 and a = — 1 represent the maximal degrees of positive and negative dependence, respectively, allowed in this family. a) If the marginals are AT(0,1) the correlation is a7r _1 (ranges from -0.318 to 0.318). b) If the marginals are uniform distributions over (0,1), the correlation is a / 3 (so it ranges from —| to — | ) . In fact, for FGM distributions with absolutely continuous marginals, the correlation coefficient between X\ and X2 cannot exceed |(Schucany et al. [194]). The FGM model can also be expressed as follows: For a e [0,1] : h2(xi,x2)

= (l-a)f1(xl)f2(x2)+af1(x1)f2(x2)

{1 + [1 - 2F1(x1)} [1 - 2^2(^2)]}

and for a S [—1,0] : fi2(xux2) = (1 -

= (l+a)f1(x1)Mx2)-af1(x1)f2(x2) M)/I(ZI)/2(Z2)

+ Hh(xi)f2(x2)

{1 - [1 - 2 ^ ^ ! ) ] [1 - 2F 2 (z 2 )]} {1 - [1 - 2F1{x1)] [1 - 2F2(x2)}} ,

where fi = —a G [0,1]. This representation naturally suggests the following generalization. Define the class of densities (De la Horra and Fernandez (1995)):

r = r+ur-,

117

Iterations

where T + = {f(xi,x2)

= (1 - a)fI(x1,x2)

+ af+(xux2),a

£ [0,1]} ,

T~ = {f(xi,x2)

= (1 -a.)fi{x±,x2)

+ af~(x1,x2),a

£ [0,1]},

fi(xix2) = fi(xi)f2(x2) is the density obtained under independence and f+(xi,x2) and f~(xi,x2) are fixed densities with marginals fi(xi) and 72(^2), representing some degree of positive and negative dependence, respectively. Obviously, the FGM model is a special case of this family, with f+(xi,x2)

= h(xi)h(x2)

f-(xi,x2)

= / i ( z i ) / 2 ( z 2 ) {1 - [1 - 2*1(11)] [1 - 2F2(x2)}} .

{1 + [1 - 2F1(x1)] [1 -

2F2(x2)}}

and

The class r + is quite similar to the class of distributions considered in De la Horra and Ruiz-Rivas (1988). However F12(x1,x2)

= F1{x1)F2{x2)[l

+ aSl{x^)S2{x^)}

0 <
is not a proper distribution with absolutely continuous marginals. Indeed its density has negative values for some pairs (xi,x2).

5.4

Iterations

From F{xux2)=F1{x1)F2(x2)[l

+

aS1(x{)S2{x2)\

one can construct the following generalization F = F i F 2 { l + a i 5 i 5 2 + a2F1F2S1S2

+ ... + ak ( F ^ W

(S^ft+i]}

k

= F,F2 + $ > , - ( F i F 2 ) W + 1 ( S i S 2 ) [ 4 + i ] .

(5.4)

118

Farlie-Gumbel-Morgenstern

Models of Dependence

Evidently, F still has the marginals Fi and F2. We shall call (5.4) the "k-1 fold iteration". In particular, the bivariate FGM with a single iteration (k = 2) can be written as F = F1F2^aF1F2SlS2+p{FlF2)2S1S2.

(5.5)

Examples of correlation values for the distribution (5.5) are : For Uniform marginals: corr(X,Y)

= ^ + ^.

For N o r m a l marginals corr{X,Y)

= - + -£-. 7T

4-7T

The ranges of parameters for iterated FGM with absolutely continuous marginals are : |a| < 1, \a + /3\< - 1 , 0 < 2 _ 1 {3 - a + (9 - 6a - 3 a 2 ) H (and not |/3| < 1 as it was initially assumed). In this case p = corr(X,Y) yields maximal value 6 _1 (132 — 1) = 0.43426 for uniform marginals and (2TT)~1 (13= - 1) = 0.41469 for normal marginals. Equivalently, for uniform marginals F(xi,x2)

= xix2 {1 + a(l - xi)(l - x2) + j3xix2(l - : r i ) ( l - ^ 2 ) } , 0 < x,y < 1

and f{xux2)

= l + a(l-2xi)(l-2:r2)+/?a;iZ2(2-3a:i)(2-3a;2), 0<x,y<

1.

Zheng and Klein (1999) study the FGM copula : C(u,v) = uv(l + o ( l - «)(1 - 1/)), - 1 < a < 1. which is a polynomial not Archimedean copula (as indicated in Chapter 4) and its iteration C(u,v) =uv + E ^ M

1

/ ^ ! - u)(l - u ) ] ( j + 1 ) / 2 , - 1 < OLJ < 1.

Dependence

119

Properties

They are of particular interest for self-consistent estimators of survival function S(t) and the reliability R(t). See Huang and Kotz (1984) for more details.

5.5

Dependence Properties

The FGM distribution is LRD (its density is TP 2 ) for 0 < a < 1, hence it is also PRD in X and Y, associated, RTI, LTD and PQD. Its density is RR 2 if — 1 < a < 0 . Indeed given f{x,y) = f(x)f(y) [1 + a{\ - 2Fx{x)} {1 - 2F2(y)}} for xi < X2 and y\ < y2, we have :

f{xi,yi)f(x2,y2) - f{xi,y2)f(x2,yi) =

4af(Xl)f(x2)f(yi)f(y2)(F(x2)

- F(Xl))(F(y2)

-

F(Vl))

which is clearly > 0 for 0 < a < 1 and negative if a is negative. 5.6

A Class of n-variate FGM Distributions

We generalize the FGM system of distributions with specified marginal distributions for the n-variate case using the following formula for a cumulative distribution function:

F(xi,...,xn)

= s=2Ki1<..,
j= l

3=1

(5.6) where the total number of parameters ai li ... i i s is card2<j
Fx(x) = l[Fj{xJ)[l+ 3=1

+

E 31 <J2 <33

J2

aj^Sj^S^)

3l<32 a

JlJ233Sh(Xl)SJ2(X2)SJ3{x3)

+ , ...,

120

Farlie-Gumbel-Morgenstern

Models of Dependence

(5.7)

+0112...n EI Sj{Xj)}.

As an example, an explicit form for the three-dimensional FGM system is : F(xux2,x3)

= Fi(Xl)F2(x2)F3(x3)(l

+ au(l - Fi(a:i))(l - F2{x2))

+ a 1 3 ( l - *i(a:i))(l - F3{x3)) + o 2 3 (l - F2(x2))(l

-

F3(x3))

r

+Qi2 3 (l - i i(^i))(l - F 2 (:c 2 ))(l - F3(x3))).

(5.8)

More details are presented in Section 5.6.1. Fj(xj) denotes the cumulative distribution function of the specified marginal distributions, j = 1,2, ...,n. The probability density functions of this generalized FGM family may be written as

J\Xl,

•••, Xn)

1 +

E

E

aiu...,i,l[(l-2Fij(xij))

s=2 l < » i < . . . < i , < n

j=l

3=1

(5.9) The marginal distributions related to distributions given by (5.9) have probability density functions of a similar form

f(xkl,...,xkm)

! +£

E

ctri,...,r.]l(l-2Fr.(xri))

s=2k1
1 < kj < kj+i

j=l

< Ti-

lt is easy to observe that parameters in (5.6)-(5.9) must satisfy the following conditions

A Class of n-variate

1 + 53

where

n,...,r, 1 1 ^ - °

fc!
121

Distributions

a

E

s=2

FGM

ICr^l < 1

j=l

for 1 < k\ < km < n and successively for m = 2, 3, ...,n. The conditional distributions connected with distributions (5.9) have conditional density functions of form:

J\xki

> •••> 3-fcj | -Efci+i) •••) xkn

) T -1

i+E

E

s=2

«fci

^n(i-2^K-))

fci+1
EE^i

j=l

r. n ( l - 2 F , , ( z , 3 ) )

s=2 G

•X[fki{xkj)

(5.10)

j=l

where G = {ki < n < ... < rs < kn,(r1,r2,...,rs) with s = 2, ...,n — i.

^ (hi,h2,

...,hs)}

The conditional distributions for the FGM family are conveniently defined by the survival functions :

Si I Z i / P | X , > Xj

S i ( x i ) l 1+ J^ ajJlFj(xj)FJ1(xil)\

J^

j<31

aJJ1J2Fj(xj)FJ1(xJ1)FJ2(xJ2)

3<J1<32

j

D

where D

=

1

+ E

a

hh(l

Ji
+a12...nl\(l-2Fj(xj)). J'=I

~ 2 ^ 1 f e 1 ) ) ( l - 2Fh(xh))

+ ..

122

5.6.1

Farlie-Gumbel-Morgenstern

Models of Dependence

A class of bivariate FGM distributions marginal distributions

with

Weibull

Bivariate FGM with exponential marginals is well known as bivariate Gumbel exponential distribution, widely discussed in the literature. Here we shall study its extension for Weibull marginals. This distribution was also described by Johnson and Kotz (1975) and Lee (1977). By substituting

F i (i i ) = l - e x p f - ^

J ,

i = l,2

into Eq.(5.1), we obtain a class of bivariate distributions with Weibull marginals

F{xux2)

= { 1 - exp | - ( ^ J

j

.01

1+

*[ ""[-{t)

1 - exp f -

/

( g

\ 02

-{?,))}

("»

where Xi > 0,ft > 0 (a curve shape parameter), 6; > 0 (a scale parameter), i = l,2 and a dependence parameter a 6 (—1,1). From Eq.(5.11) it follows that the density functions of a class of bivariate distributions with Weibull marginal distributions are of form :

,/

N

ft #2 (Xl

(5.12) where x{ > 0, ft > 0, 6j > 0 ; i = 1,2, a e (-1,1). The distribution functions (5.11) and the density functions (5.12) are expressed in the forms which is symmetric in the variables, it is therefore sufficient to consider a conditional distribution of Xi, given

A Class of n-variate

FGM

Distributions

123

i — 1,2, when the joint density of X\, X2 is described by (5.12). Hence the conditional density function of Xi, given Xj = Xj, i ^ j , i = 1, 2 is ft\\

/(*,!*,) = J l + a

l-2exP

x

- ( g )

/

1

/ „ \ ftiN

/

l-2exP

- ( g )

fl / N/3"-1 / / \ Pi Pi ( ^C% \ exp I f •&%

vAVi

"U"

The regression of Xi on Xj is given by

E{Xi\Xj=Xj)=biY{l

x < l + a l - 2 exp

1

+ j)

(1 2 /ft)

(*)*)) - " }-

<513)

'

* 7^ i i *>i — 1)2. The conditional expectation E (Xi | X,- = Xj) increases if a > 0 (decreases if a < 0) starting from fc^l + 7J7) [l - a ( l - 2"1/'3*)] to 6^(1 + 7J-) [l + a ( l - 2- 1 / f t )] provided that the variable Xj increases. From (5.13), denoting E (Xi | Xj = Xj) = Ea (Xi \ Xj), it is easy to obtain the following relation : i [E.a (Xi I

XJ)

+ Ea (Xi I Xj)] = hT(l + j)

i^ j , i,j = 1,2.

This relationship indicates the symmetry between Ea (Xi \ Xj) and E-a (Xi I Xj). Conditional moments of order k possess a similar property :

\ [E^a (X? I XJ) + Ea (X* I XJ)]

= &*r(l + A).

The conditional variance of Xi , given Xj = Xj, is given by

Var (Xi I x3) = btT{l+j)

I 1 + a I 1 - 2exp I - ( ^ )

) ) (1 - 2 " 2 ^ )

124

Farlie-Gumbel-Morgenstern

Models of Dependence

pr

-^r2(l + - W l + all-2exp

* ^hhj

(1_2-Vft)

= 1,2.

It is easy to verify that

E{X?X?)

+ ^-) [l + a(l - 2- Ul /' 3l )(l - 2-" 2 ^ 2 )l

= b^b^T{l + ^)T(l Pi

pi

L

J

(ui,u 2 ) G i?5-. In particular, using this expression the covariance between X\ and X 2 is found to be Cov(XuX2)

= afc 1 6 2 r(l + - i ) r ( l + -j-)(l - 2 - 1 ^ ' ) ( l - 2- 1 / / 3 2 ) Pi

P2

and the correlation coefficient is (i _ 2 - i / f t ) ( i - 2 - 1 / f t ) r ( i + ^ ) r ( i + ^ ) p = a-

([r(i + ^)-r 2 (i + ^)][r(i + ^)-r 2 (i + i)]) 1/2

Note that the least squares linear regression of Xi on Xj has as its equation 6,r(i + f )r(i + f )(i - 2- 1 /^i)(i - 2-vft) E(Xt I X,) = a — SlLl ^ —, -Xj bi r ( i + ^ ) - r 2 ( i + i )

r 2 ( l + J-)(l - 2- 1 /ft)(i _ 2-V&) +6,T(1 + - W 1 - a

A {

a linear function in Xj.

[r(i + ^ ) _ r 2 ( i + ^ )

A Class of n-variate

5.6.2

A class of three-variate marginal distributions

FGM

125

Distributions

distributions

with

Weibull

As it was mentioned above, the FGM system of bivariate distributions with specified marginal distributions may be extended in case of three random variables as follows

F(XI,X2,X3)

=

1 ++

^^2

S x

Ot arsrsSSr(Xr)S + ai23 Y[ s(xs) a)+ai23\[Sj(Xj) r(xr)Ss(x

j( j)

JJ.Fj(a:j)

Kr<)<3

(5.14)

where SJ(XJ) = 1 — Fi(xi) and Fi{x{) is a specified cumulative distribution function, i = 1,2,3, and the parameters aTS € (—1,1), 1 < r < s < 3. The density functions describing the system of three-variate distributions with specified marginal distributions Fi(xi), i = 1,2,3 are of form

1+

/(zi,^.^)

£

ars{l - 2Fr(xr))(l

-

2Fs{xs))

l
+

(5.15)

aiaalKl-ZFj-fo)) x n

fi(xj).

The marginal bivariate distributions associated with distributions having cumulative distribution functions (5.14) or probability density functions (5.15) are given by Fij(xi, Xj) = Fi(xi)Fj(xj)

fijixi, Xj) = fi{xi)fj(xj)

[1 + aijSi(xi)Sj(xj)]

[I + aij(l - 2Fi(xi))(l

, 1< i <j < 3,

-

2FJ{XJ))}

,l
A trivial case of independent random variables implies a^- = 0, 1 < i < j < 3, CK123 = 0. The conditional distributions connected with distributions (5.15) may be described by conditional probability density functions

126

Farlie-Gumbel-Morgenstern

f(xi\Xj,xk)

= x

Models of Dependence

fi(xi){l + [l + ajk(l-2Fj{xj))(l-2Fk(xk))}-1 [(1 - 2Fi{Xi)) ( a y ( l - 2Fi(xi)) + aik{\ - 2Fk(xk)))

+

a123f[(l-2Fr(xr))]} r=l

where i ^ j , i ^ k,j < k;i,j,k

G {1,2,3}.

/ \X{, Xj I Xk J

—

Ji\xi)Jj\xj)

1+

Y,

ars(l

~ 2Fr(xr))(l

-

2Fs{xs))

Kr<s<3

+a123f[{l-2Fr(xr)) r=l

where j > i, i ^ k, j ± k ; i,j, k G {1,2,3}. Using formulas (5.14)-(5.15) to obtain a system of three-variate distributions with Weibull marginal distributions we set Fi(xi) = l - e x p ( - : r f )

fi(xi) = ftzf-1 e x p ( - z f )

x{ > O.ft > 0. (5.16) Here it is assumed without loss of generality, that bi = 1, i — 1,2,3, for the cumulative distribution functions and probability distribution functions of a class of three-variate distributions with Weibull marginal distributions

F(xi,x2,x3)

=

1+

or

X]

"rs exp(-xf" - x f ' ) + a 1 2 3 e x p ( - ] P a ^ ' ' )

Kr<s<3

xn(l-exp(-xf)) j=i

r=l

(5.17)

A Class of n-variate

f(xi,x2,X3)

1+

=

FGM

Yl

Distributions

127

ar,(l-2exp(-x£"))(l-2exp(-a:?'))

l
-a123n(l-2exp(-xf))) j=i

f[Pjxf

'expi-xf)

ars G (-1,1)

J=I

(5.18)

(^1,2:2,^3) e # + .

The conditional probability density functions for this class of distributions (5.18) are

f(Xi

I Xj,xk)

A> i j -(l - 2 e x p ( - aJi> {1 + [(1 - 2exp(-a:*))(a :f))

=

3

+aik{l

- 2 e x p ( - i f * ) ) - a123 J J ( 1 - 2 e x p ( - i ^ ) ) ] r=l

1 + ajk(l

-2exp{-x /)),& p

+ (1 - 2exp(-a£ f c ))

x/3ixf-1exp(-a;f) where i ^j,i^k;j<

k;i,j,k

6 {1,2,3}.

J" ^x^ j 2?j J ^ / c j

= [1+

£

ar,(l-2exp(-af"))(l-2exp(-a^'))

l
3

-«123 I I ( 1 - 2 e x p ( - ^ ) ) ] r=l

where i <j,i^k;j

^k;i,j,k

<E {1,2,3}.

~[[l3r4<-1exp(-4'-)

}

128

Farlie-Gumbel-Morgenstern

Models of Dependence

The regression of Xi on X,, Xk is

E(Xi | Xj,xk)

x ( l - 2-1'^){{aij{l

- 2 e x p ( - x f ) ) + aik(l

-a123f[(l-2exp(-x^))}}T(l .

+ (1 - 2exp(-zf f c ))]- 1

= {1 + [1 + a i f c (l - 2exp(-xf))

+ ~)

i

iyLj,i^k;j<

-2exp(-x%k))

Pi

k;i,j,k

€ {1,2,3}.

Taking into account the expression

E{X^X%2X%3)

=

[1+

Yl

ay(l-2-Ui//3i)(l-2~""i/ft')

l
u

3

r i+ - a 123 n(i-2- ^)]n ( 0r' ?) r=l (Ui,U2,U3) G # + •

One easily obtains the elements of the covariance matrix (i,j = 1,2,3). Namely

Cov(Xi,Xj)

=

{Cov(Xi,Xj)};

a y r ( l + J r ) r ( l + i ) ( l - 2- 1 /ft)(l - 2-Vft)

r(i

2

+

A)-r (i + i )

i jL j « =

j

(5.19) It is easy to verify that Det{Cov(Xi,Xj)} correlation matrix {pij} are of form

^ 0. The entries of the

(i-2-1^)(i-2-1^-)r(i+^-)r(i+^)

aU Pij = *

-rj2 ([r(i+^)-r2(i+i)][r(i+^)-r2(i+i)]y'_

i^j i = j

. (5.20)

A Class of n-variate

FGM

129

Distributions

Numerical calculations for the values of pij are straightforward. 5.6.3

FGM n-variate

distributions

with Weibull

marginals

The construction of a class of n-variable distributions with Weibull marginal distributions is carried out by inserting (5.16) into (5.6) or (5.9). Hence this class of n-variable distributions with Weibull marginal distributions is described by cumulative distribution functions

x

-T \X\ , . . . , Xn J

+E

Hu

"ii,...,i. II exp ( -:E ^)

E

s=2 l
j=l

(5.21)

nCl-expHrf)) or by probability density functions

JyXi,...,

xn)

—

1+E

E

(-l)"«*i

s=2 1
&<

w I l ( 1 - 2exp(-xf)) j=l

(5.22)

x n ^ ' ^ l - e x p ^ ) ) .

The product moments oiXi,X-z,..., Xn possessing the distribution (5.21) can be obtained by using the functions

E(X?...X?)

i+E

E

s=2 l < i i < . . . < i 3 < n ^

r

T

l

+

^

V

(-i)'«
130

Farlie-Gumbel-Morgenstern

Models of Dependence

The entries of the covariance matrix are given by (5.19) and elements of the correlation matrix are like (5.20). The conditional probability density functions follows from (5.16) and (5.22)

f(xkl,...,xki

|

xki+1,...,xkn)

= {! + [!+ E* s=2ki

E

s

X[E E(-l) «n,...,. s s=2 G

x

(-l)'a*x,..,A. n ( l - 2 e x P ( - ^ ) ) ] - 1

+ 1
j=l

fl(l-2eXp(-a;^))]} j=l

n^xf-^l-exp^))

G = {kx < n < ...
^ (h1,h2,...,hs);s

=

2,...,n-i}.

Hence, the regression of Xkl on Xk2, ...,Xkn is

E(Xkl

|

Xk2,...,Xkn)

= 0 + [l+ E1

E

(-l)'afcl

s=2 ki + i
x[E E(-l) «n

h.

fl(l-2exP(-^))]-1 j=l

r.(l - 2 - ^ ) ft (1 - 2 e x p ( - x ^ ) ) ] } r ( l + £-) ,

where as above G = {hi < n < ...
5.7

^ (h1,h2,...,hs);s

=

2,...,n-i}.

Further Extensions

In this section, we shall discuss a number of extensions of FGM distributions with uniform marginals designed to increase the maximal value of the correlation coefficient. We thus discuss a number of polynomial copulas.

Further

5.7.1

Huang and Kotz

131

Extensions

extensions

1. In this case, the joint distribution function of bivariate variables (X, Y) is:

Fa(x,y)

= xy[l + a(l-xp)(l-yp)},p>0,

0<x,y
(5.23)

And the pdf is fa(x,y)

= 1 + a ( l - (1 + p)xp)(l - (1 + p)yp).

The admissible range for a in this case is given by -(max{l,p})~


The range of corr(X, Y) = , ^ ° L is : - 3 ( p + 2)" 2 m i n { l , p 2 } < p <

^ - ^ ;

i.e. for p = 2, p m a x = | , and for p = 1 p m i n = - 3 / 1 6 . The introduction of parameter p allows us to increase the maximal correlation for the FGM-ditribution with uniform marginals. See Fig. 5.1.

2. Alternative extension Another extension of the bivariate FGM with the uniform marginals is given by

Fa(x, y) =xy[l

+ a ( l - x)p(l - y)p], p > 0,0 < x, y < 1.

(5.24)

Here the p.d.f. is fa(x,y)

= 1 + a ( l - x)p-l(l

- yy1^

and the admissible range of a is (for p > 1): -1 < a < ,

p+ l x

- (1 + p)x)(l - (1 + p)y)

132

Fig. 5.1 (5.23).

Farlie-Gumbel-Morgenstern

Models of Dependence

Bounds on correlations p as a function of parameter p for the distribution

The range is empty for p < 1. The corr(X,Y) between -12

(p + 2 ) ( p + l )

-p-12

= 12a ((p+2Wp+i))

^T2P

var es

i

•

i.e. for p = 1.877, p m a x = 0.3912, p m i n = -0.333. Hence, in this case, the maximal positive correlation is even higher than the one attained in the first extension of Huang and Kotz given by Eq. (5.23). See Fig. 5.2.

5.7.2

Sarmanov's

extension

The density of the joint distribution of the variables X and Y is given here by the expression : f(x,y)

=

1 + 30(2* - l)(2y - 1) + - a 2 [3(2s - l ) 2 - l] [3(2y - l ) 2 - l] 0<x,y

<1

(5.25)

Further

Fig. 5.2 (5.24).

133

Extensions

Bounds on correlations p as a function of parameter p for the distribution

and distribution function is F{x,y)

=

xy{l

+ 3a(l-x)(l-y)

0<x,y

+

5a2(l-x)(l-2x){l-y)(l-2y)}

< 1.

I.O. Sarmanov (1974) makes brief mention of the distribution with density (5.25), obtained using the first two Legendre polynomials. If a satisfies the condition , • V7 \a\ < — ~ 0.55 5 then the function (5.25) is nonnegative in square [0,1] x [0,1] and is a genuine probability distribution. Moreover a is also the coefficient of correlation between X and Y. If we substitute arbitrary distribution functions X and Y, for example, the incomplete gamma functions into (5.25)

= f ] S T i ) / ^ • ^ y = f(STT)/''°«"•dy we shall obtain a new type of two-dimensional gamma correlation surfaces. Further investigations of this family which allows for a rather high correla-

134

Farlie-Gumbel-Morgenstern

Models of

Dependence

tion between the variables seem to be desirable. 5.7.3

Sarmanov-Lee

extension

The class of a bivariate distributions introduced by O.V. Sarmanov (1966) encompasses the FGM family. Specifically, let / and g be univariate p.d.f.'s. Let ipi(t),i = 1,2 be bounded nonconstant functions such that oo

oo

/ f(t)il)1(i)dt = Oznd — oo

/ g(t)ip2(t)dt = 0. —oo

Define a bivariate density function ha(x,y)

= f{x)g(y){l

+ atl>l(x)rh(y)}

(5-26)

with specified marginals f(x) and g(x), where a is a real number satisfying the condition that 1 + atpi(x)ip2(y) > 0 for all x and y. The family with joint p.d.f. (5.26) is sometimes called the Sarmanov family of bivariate distributions. This construction is identical with Riischendorf construction (see Chapter 4 Section 4.5.1), who possibly was not aware of this work, which appeared in a mathematical Russian Journal. Lee (1996) considers "kernels" of type ipi(x) = x — Hi,ip2{y) = y — 1^2, where /ii = E(X), fi2 = E(Y), and shows that the range of correlation coefficients for this family of distributions is determined by both the marginal distributions and their mixing functions ip, (i = 1,2) and may therefore be wider than the one for the FGM distributions. With Lee's model we are thus returning to polynomial situation. The range of a in this case is:

f-L

zi

max I , Jr— r-) < a < min(V/X1M2 ( 1 - M i ) ( l - M 2 j M1U-M2)

Ml-Mi), (5.27)

and correlation coefficient is given by P = ° ^ , ax cry

(5-28)

where vi = Jxip1{x)f(x)dx, u2 = /yip2(y)g(y)dy and ax = \/Var(X), ay — y/Var(Y). However for the uniform [0,1] marginals it follows from (5.27) and (5.28) that -Ka<

1

Further

135

Extensions

and 1 1 3 - ^ - 3 ' which is the same range as for the classical FGM. Actually Lee's model leads to the original FGM family. 5.7.4

Bairamov-Kotz

extensions

Bairamov and Kotz (2000a) investigate a two-parameter extension of the FGM family adding an additional parameter to the Huang-Kotz distribution discussed in Section 5.7.1. 1. The distribution function of (X, Y) is given by : Fa,a,b(*> V) =xy[l

+ a ( l - x a ) 6 (l - ya)b] , a > 0, b > 1,0 < x, y < 1,

with the pdf /a,a,b(z, y) = 1 + a ( l - x")"-1^

- yaf-1

[1 - xa(l + ab)] [1 - ya(l + ab)}.

The admissible range of a is: for b > 1, • m i n < 1,

1 .ab + l> 6 -i'

1 a b + 1 ;,_!


v

V 6-1

l ^ b - l '

and for b = 1, the quantity inside the square bracket is to be taken as - . The maximal and minimal values of corr(X,Y) are within the range -12i 2 (6,a)min^ 1,

1 ab+1 abK b - 1 '

2^ 6-1


= 12a

b r(t)r(o/2) ab+2 r ( b + j )

1 + ab 12t2(b,a)a \a(b — 1)

6-1

(5.29) where t(x,y) =

,

x 1

t

L , and T(a;) = f t ~ e~ dt— is the gamma func-

tion. For a = 2.8968 and b - 1.4908, we have p m a x = 0.5015. And for a = 2 and 6 = 1.5, p m i n = -0.48. An alternative extension is a generalization of Huang-Kotz family : 2. Here the distribution function of (X, Y) is Fp,q,n;a(X> V) = * V i1 + a(l ~ S T ( 1 - VT}

, P, 9 > 1, U > 1,

136

Farlie-Gumbel-Morgenstern

Models of Dependence

0<x,y
(5.30)

with marginals F\ {x) = xp, F2 (y) = yp, 0 < x, y < 1. The pdf is Jp,q,n;a\X, — ™ P - I „pPx- I

y-

2 {pr„2 + a(l - z T

- 1

[(P -

y)

Z?(P

+ «")] (1 - v T _ 1 [(p-2/ 9 (p + ?n)]}.

(5.31)

The admissible range for a: min { 1, —

p + gn ?(« - 1)

2(n-l)'

< a < -

p + qn q(n - 1)

n-l

and the admissible range for correlation coefficient is -a{n,p,q)

I 1,— P min <

p + gn «(n - 1)

2(n-l)'


p + qn q(n - 1)

n-l

where 1

a( n,p,q) = f

/

2

(p + 2 ) ( p + l ) V

g V

_L 1

«

For p = 1, q = 2 the maximal negative correlation p m j n = —0.4794 is attained at n = 1.495. For p = 0.001, q = 1.5 the maximal positive correlation p m a x = 0.6122 is attained at n = 1.379. It should however be noted that (5.30) is not a copula. 5.7.5

Lai and Xie

extension

Lai and Xie (2000) consider a bivariate function C(u,v) =uv + w(u,v)=uv

+ aubvb{l-u)a(l-v)a,

a,b>l,

(5.32)

(which is, indeed a particular case of the Riischendorf construction) and have shown that (5.32) is PQD for 0 < a < 1. Bairamov and Kotz (2000b) observe that (5.32) is a bivariate distribution function for a over a wider range satisfying

[B+(a,b)}

[B~(a,b)]

2 I


B+(a,b)B-(a,bY

Further

137

Extensions

and possessing the PQD property for a satisfying 1 B+{a,b)B-(a,b)'

0 < Q <

where T 6-1

B+(a,b)

b(a + b-l)

=

+ sjab{a + 6 - 1 )

(a + b) (a + b - 1) b(a + b - 1) + ^ab(a + 6 - 1 ) (o + b) (a + 6 - 1)

1,

a-l

,. 6(a + 6 - 1) + A/a6(a + 6 - 1 ) (a + 6)(a + 6 - l )

( a + 6)

, &

(5.33)

and

B~(a,b)

=

b(a + 6 - 1) - y/ab(a + (a + 6) (a + 6 - 1)

b-l)

6-1

6(a + 6 - 1) - y/ab(a + 6 - 1 ) (a + 6) (a + 6 - 1) ,

, ,.6(a + 6 - l ) - y/ab{a + (a + 6)(a + 6 - l )

{a + b)

a-l

b-l) *

(5.34)

For example letting a = 5 and 6 = 5 we have from (5.33) and (5.34) ~ B+(a b)B-(a b) ~ 6-053 x 104 which is by far larger than 1. Further investigations of the Lai and Xie family could be fruitful. 5.7.6

Bairamov-Kotz-Bekgi

generalization

Our final generalization of the FGM family involves four parameters. The distribution function of (X, Y) is F(x,y)

=

xy{l + a(l-x^)qi(l-yP2)q2} 0<x,y
pup2>l,quq2>l (5.35)

138

Farlie-Gumbel-Morgenstern Models of Dependence

The pdf is f(x,y)

x^)"1-1

= l + a(l-

[1 - (1 + Piqi)

x(l-^)92-1[l-(l+p2g2)2/P=]

x^}

0<*,y
(5.36)

The admissible range of a is

P1P2 \P\ (qi - I) J

V1"1 '

j 1 f l+piqi < mm' '

Pi\Pi(9i-l)/

The admissible range of corr{X,Y) is

\P2 (92 - 1)

N92-1'

1 ( l+p2q2 ' ' P2 \ P 2 (
= p = C°^p

=

— 12* (
P1P2 \Pi (qi - 1) /

12at(q1,p1)t(q2,p2)

\P2(92-1)


12t{ql,p1)t(q2,p2)mmi

— [— — [Pi VPiWi - 1 ) /

,— — — P2\P2(92-1)/

}, J

where £(.,.) is defined in section 5.7.4 above Eq. (5.29). Clearly (5.35) is the most general form of modified FGM distributions. It is of interest whether (5.35) satisfies T P 2 property for all pi,p2 > 1, <7i>92 > 1 or not. The answer is negative and contained in the following example. The condition allowing to be TP 2 is that for all x\ < x2 and y\ < y2 f{xi,yi)f(x2,y2)

> 0.

(5.37)

For the distribution (5.36) denoting a{x;p,q) = [1 - xP]9'1^ (5.37) can be written as follows:

- (1 + pq)xp]

Xiyi{l

- f{xi,y2)f{x2,y1)

+ aa(xi;pi,qi)a(yi;p2,q2)}x2y2{l

+

aa{x2\pi,qi)a(y2;p2,q2)}

Further

-xiy2

{1 + aa(xi;pi,qi)a(y2;p2,q2)}-xiyi Xiyiyiy2<x[a(xi;pi,qi)a(y1;p2,q2) -a(x1;p1,qi)a(y2;p2,q2) x\y\yiy2a

139

Extensions

{1 + aa{x2;pi,qi)a(yi;p2,q2)} +

a(x2;pi,q1)a(y2;p2,q2)

- a(x2;p1,qi)a(yi;p2,q2)]

{a(xx;px, qi)[a(yi;p2, q2) -

-a(x2;p1,qi)[a{yi;p2,q2)

>0

> 0

a{y2;p2,92)]

- a{y2;p2,q2)]}

>0

or xiyiyiy2a[a(x1;pi,q1)

- a(x 2 ;Pi,9i)]

xKj/i;P2,92) -a(2/2;P2,?2)] > 0.

(5.38)

But if /(a;) = 1, 0 < x < 1, i.e. if the marginals are uniform distributed on [0,1], for a = 1, pi — 2, P2 = 2, qi = 2, q2 = 2 and zi = 0.1, x2 = 0.2, yi = 0.7, xi — 0.9 from (5.38) one has f(x1,y1)f(x2,y2)

- f(x1,y2)f(x2,yi)

= -0.0276 < 0,

i.e. X and Y are not TP 2 , whenever classical FGM is T P 2 by the Lemma 1.1. See Bairamov et al. (2000) for further details. 5.7.7

Concomitants

of order

statistics

Let (Xi,Yi), i = 1,2, ...,n be a random sample from an absolutely continuous bivariate population (X, Y) with a d.f. FX,Y (x,y)- Let Xr:n denote the rth order statistics of the X sample values. Denote by Y[T:n] the Y value associated with Xr:n. We call yj r:n ] the concomitant of the rth order statistic. For more details the reader is referred to the review articles of Bhattacharya (1984) and David (1993). More recently Balasubramanian and Beg (1997), (1998) have studied concomitants in FGM family with exponential marginals. Denote probability density function (p.d.f.) of Yjr:„] by 9[r:n] iv)• It is known that +00

5[r:n] (y)=

/ — 00

/ (V I X) fr:n

(x)

dx

Farlie-Gumbel-Morgenstern

140

Models of Dependence

where / (y \ x) is the conditional density function of Y, given X and fr-n (x) is the p.d.f. of XT:n (see David (1981)). The most important use of concomitants arises in selection procedures when r (1 < r < n) individuals are chosen on the basis of their X values. Then the corresponding Y values represents performance on an associated characteristic. The distribution and recurrence relation between moments of concomitants in the classical bivariate FGM distribution with uniform marginals are given below. The d.f. and p.d.f. of Y"[r:n] are respectively G[r:n] (y) = V

ll + a 1

71 + 1

(1 - y ) } , o < y < 1

(5.39)

and 9[r:n] («/) = 1 + a

71 + 1

(1 - 2y) •

(5.40)

See Bairamov and Bek$i (1999) for additional details. For the generalized distribution (5.35), we have the following result : Theorem, a) The kth moment of Y[r:n] is (*) _

1

/V:n] - jfe + 1

+

aC{k;p2,q2)K(r,n,p1,qi).

Thus the expected value and variance of Y[r:n] are E{Y[r:n]} = ~ + <*
=— +

aK(r,n,p1,q1)

- C(l;p2,ga) [1 + aC{l;p2,q2)K{r,n,puqi)]}

,

respectively. The moment generating function of Y[r:n] is

M[r:n] (t) = E{etYw}

=

e

-^+aK(r,n,Pl,gi)

= j e'^[r:n] (y) dy

£ (-1)'' ( * . J [S(jp2) - (1 +p2q2)S((j + l)p2)}

FGM

141

Sequences

Here B(a, b) is the beta function : B{a, b) = J* ta~l{l -tf^dt, 0 and

^».p.«) = D - 1 ) ' ( ' < 1 ) i ( ^ r + x[B(ip+r,n-r

+ l)-(l+pq)B((i

+ l)p+r,n-r

a > 0, b>

l)

+ l)] , (1 < r < n,p, 1 < q)

C(k;p,q) = i [ 5 ( * ± i l 9 ) - (1 +pq)B

(*±I,g)

(fc = 0,l,2,...; l < p , 9)

5(a) =

5.8

T

(-l) f c a! fc=0 k t (a-k)\

+ (-1)

0+1 fa+l

FGM Sequences (1) Limiting distribution Define a FGM distribution F in Rn, for n > 1 with respect to the given univariate distributions F;, i < n, by

F(a:i

) = 1 1 ^ ^ ) \1 i=l

^

+

5Z

a(j,k)Sj(xj)Sk(xk)

Kj'
(5.41) for all vectors x = (xx,X2,---,xn) € R , where the n(n —1)/2 terms a(j, A;) are suitable constants, such that F is a distribution function, where SJ(XJ) = 1 — FJ(XJ). This is a particular case of a general FGM distribution, discussed in Section 5.6, where a(j,k,l) = 0, a(j, k, l,m) = 0 and so on. The univariate marginals of F are the F{. The constants a(j,k) are admissible if the 2" inequalities n

1+ Yl

a(j,k)^k>0

l<j
hold, for all £j = —Mj or 1 — rrij, where Mj = sup {Fj(x), - 0 0 < x < 00} \ [0,1]

(5.42)

142

Farlie- Gumbel-Morgenstern

Models of

Dependence

and rtij = inf {Fj(x), - c o < x < 00} \ [0,1] . If Fj is absolutely continuous, then Mj = 1 and m.j = 0, hence £.,• = ± 1 . Hashorva and Hiisler (1999) define a FGM random sequence {Xj,j > 1} with univariate marginals Xi ~ Ft, i > 1, and a symmetric function a(.,.) (a(j, k) — a(k,j)) such that the joint distribution of X^,Xi2, ...,Xin is given by the FGM distribution

F

ii,i2,-,in(x)

f

=n^*^) \1 i=l

+

5Z

y

aiij^^Si^x^Si^Xk)

l<j
(5.43) The parameters a (.,.) are admissible if for every n > 1 and {ii, %2, •••,in] the inequalities 1+

]L

a

(^.*fc)^^>°

l<j
hold for all £^. The FGM sequence is stationary iff the univariate marginals are all equal, Fi = Fui>

1,

and the parameters a(j, k) depend on j , k only through their difference, i.e.: a

tii k) = a{j — k) for all j ^ k.

Let also limn-^oo sup-_ f c > n \a(j, k)\ —> 0 Consider partial maxima Mn = maxi<„ Xi, n > 1. Let {Xi, i > 1} be a sequence of random variables JQ whose joint finite dimensional distributions are FGM. Hashorva and Hiisler (1999) derived the limiting distribution of the maxima with respect to some suitable normalization un(x). They consider an approximation to P {Mn < un(x)}. In the case under consideration the dependence between the random variables Xi is not very strong and the approximation

P{Mn < un(x)}

tt]JP{Xi<

un{x)} =

Y[Fi(un(x))

is valid. Suppose that the normalization un(x) satisfies lim supFi(un(x)) ™-*-°° i
=0

(5.44)

FGM

Sequences

143

for the set of a; with liminf„^oo \[i 0. This condition is essential for general results in the general case of nonidentically distributed Xj. More precisely, suppose that {Xi, i > 1} be an FGM sequence such that (5.44) as well as sup \a(j,k)\

—¥ 0,n —>• oo

j — k>n

and lim sup yjSj(u n (a;)j < °° n—foo . ~ i
hold with respect to a normalization un(x). Then P {Mn < un(x)} - J J Fi(un(x))

-»• 0

i
as n —> oo. In addition if lim T\Fi{un(x))

n—>oo •*••*•

=G{x)

i
then P{Mn

< un(x)} A G(x)

as n -> oo. Compare with Joe [ill], page 178, who studied related limiting distributions in the context of a distribution of extreme value theory. (2) Order statistics Let X\, X2,..., Xn,... be a sequence of uniform FGM random variables such that for any n > 1 Fn{xl,x2,...,xn)

= Y[xt

> , J (5.45) where n(n — l ) / 2 terms an(j, k) are suitable constants. Since Fn is a distribution function, i=1

1+

I 1+ J2 oin(j,k)(l-xj)(l-xk) (, l<j
Y. l<j
a

nU,k)^^k>0

(5.46)

Farlie- Gumbel-Morgenstern

Models of Dependence

holds, where £, = 1 or —1. Let an(j, k) — an for all j , k. Then for any n the random variables Xi, X2, • ••jXn are exchangeable. Namely (-X»i> X{2,..., Xin) — (Xjl,Xj2,...,

Xjn).

It follows from (5.46) that in this case the admissible range for an, allowing (5.45) to be a n-variate copula, is

"(f)- a n -M' where [a] denotes the integer part of the number a. In this case the coefficients an, allowing (5.45) to be a copula converges to 0, for n -> 00, i.e. for large n the FGM sequence becomes independent. (One can call asymptotically independent.) Exchangeable FGM random variables were originally studied by Shaked (1975) [200]. Consider now the finite FGM sequence Xi,X2,.-.,Xn. Denote by X\:n < X 2: „ < ... < Xn:n the order statistics of X\,X2, • •. ,Xn. It is known that for symmetrically dependent random variables

p {xr!„ < * } = E ( - i ) ' - r ( ' : i ) (") p {x,..i < x}. For (5.45) we have Fr:n(x) = E z = r ( - l ) ' - r ( ' : 1 l ) ( ? ) ^

P(Xr:n <X)=

{l + anl-^(l-x)2},

(5.47)

since for any integer m,

P{Xm..m <x}=Xm

| l + Q r o m ( m 2 ~ 1 ) ( l - Xf J .

Let Xn+1,Xn+2,---,Xn+m be a new FGM sample fitting to the same model which is assumed to be independent of Xi, X2, •••, Xn. Then P{Xn+i

< Xr:n} = 1 — P{Xn+i

>

Xr:n}

FGM

= 1 - / P{Xn+1

Sequences

145

> Xr.n I Xn+1 =x}dx

= l-

0

P{Xr..„ < x}dx 0

= 1_X-(_l)™-r(™-A(n\(^_

~.

U - i y W \ m + l

m(m-l) "(m + l)(m+2)(m + 3 ) / '

Denote ^

vm = ^2 & • i=l

It is evident that vm is the number of Xn+i, Xn+2, •••, -XWm falling below the random threshold Xr:n. Bairamov and Eryilmaz (2000) derived the exact distribution of vm. Their result is as follows. For any integerTO> 1 and 1 < r < n P{vm

= k}

= (T)B-irr(;:l)C)[^(S+fc,m-fc+i) s=r

+an (s2(s~1'>B(s + k,m-k + 3) - s(s - l)B(s + k + 1,TO - k + 2)j +am (sk^2~^B(s

+ k,m-k + 3)- sk(m - k)B(s + k + l,m-k + 2)

+ '("-fc)( 2 "-fc-i),g( 3 +anam

+ k+2

(^s2%-1)k^B(s

, m _ k + 1))

+ k,m-k + 5)~ ^M^k(m

xB(s + k + l,m-k + 4) + ^

^

- k)

(m-k)(m-k-i)

xB(s + k + 2,m-k + 3) -s(s - 1)^~>-B(s

+ k + l,m-k + 4) + k{m - k)s{s - 1)

xB(s + k + 2,m - k + 3) _

g(s

_ i) (™-fc)(m-fc-i)3 (a

+ k + 3)jn_k

+ 2)y

146

Farlie-Gumbel-Morgenstern

k = 0,l,...,m ,

Models of Dependence

—7-r
FGM

147

Sequences

Below we present some numerical values of P {ym = k} for selected values of the parameters n, r, m, an and am. These results are quite informative. (Compare, in particular, the first and third rows.)

Table 1. Numerical values of P {ym = k} . n

r

3

1 2

m

an

am

k

P {vm = k}

0.5

-0.25

0 1 2

0.629 0.267 0.104

-0.1

-0.5

0 1 2

0.403 0.422 0.175

0.1

0.75

0 1 2 3

0.347 0.310 0.231 0.112

Chapter 6

Global Versus Local Dependence between Random Variables

6.1

Introduction

Global indices were used and continue to be used in probability theory and statistics for many years to measure the dependence between two random variables X and Y. As it was emphasized in Chapter 2 among these indices the linear correlation coefficient, Spearman's correlation coefficient on ranks, Kendall's concordance coefficient are by far the most prominent. The linear correlation coefficient is appropriate mainly for normal variables, the two others are independent of the marginal distributions of X and Y. Some authors have measured the dependence between two random variables by mutual information or relative entropy. In the case of normal variables relative entropy is connected with the absolute value of the linear correlation coefficient. These indices are defined from the moments of the distribution on the whole plane and can be zero when X and Y are not independent. One needs therefore indices which measure the dependence locally. In the case where X and Y are survival variables, one needs to identify the time of maximal association : for example the delay before the first symptom of a genetic disease by members of the same family will appear. The pairs (X, Y) and (X',Y') can have the same global measure of dependence but may possess two different distributions F and F' : a local index will allow us to compare them, and in the case of survival variables to compare their variations in time. The variations with x and y of some local indices allow us to characterize certain distributions and conversely choosing a shape of variation for an index allows us sometimes to choose an appropriate model. 149

150

Global Versus Local Dependence between Random

Variables

In the second Section we reproduce several axioms established by Renyi [179] that such a measure should satisfy. We then give several definitions of three global indexes, and their connection with concepts of dependence. This has also been studied in the papers of Genest [81], Long and Krzysztofowicz [147], and Nelsen [160]. We next provide several definitions of mutual information or relative entropy. These notions have been established by Bell [20], Joe [108], and illustrated by Kapur [121] for non-normal distributions and multivariate vectors. Starting from a characteristic property of the relative entropy for a p + q random normal vectors (X, Y), Lin [145], and later Zografos [233] have developed new measures of dependence constructed with the canonical correlation coefficient of the two vectors X and Y. The third Section defines a few local indices studied by Bjerve, Doksum and Blyth [21], [61], [25], Dabrowska [53], Clayton [46] and Oakes [163], Holland and Wang [98] and Jones [118] and their connections with certain concepts of dependence. The fourth Section explains how to obtain a non-parametric estimation of two local indices, and the properties of these estimations. Dabrowska [53] has developed these notions. We are searching for localization of maximal dependence in the fifth Section. This is illustrated by the distributions of three copulas with the same concordance coefficient, and the evolutions of three local indices for these distributions. A part of this work was originally presented in Drouet [62]. A number of results scattered in the literature are unified.

6.2

Global Measures of Dependence

6.2.1

Desirable

properties

of a measure

of

dependence

Renyi [179], [180] and Bell[20] have discussed some axioms that global measures should satisfy; these properties are also desirable for local measures. Note that among the indices presented here, some are not measures, since they may be negative. The desirable properties should be : (1) Standardization The values of an index are between 0 and 1. (2) Independence If X and Y are independent, the index should be zero. (3) Functional dependence If one variable is a function of the other, the index must be equal

151

Global Measures of Dependence

to 1. (4) Increasing property The index should increase as the dependence increases. (5) Invariance The index should be invariant with respect to a linear transformation of the variables. A stronger condition would be that the index is marginally free, i.e. the measure of the dependence is the same as the corresponding measure on the copula. (6) Symmetry If the variables are exchangeable, then the index should be symmetric. (7) Relationship with measures for ordinal variables. If the index is defined for both ordinal and continuous variables, there should be a close relationship between the two measures. (8) Interpretability. This is a very delicate and intangible property. Roughly speaking, it means that the numerical value of this index can be translated into a qualitative meaningful measure. 6.2.2

Covariance,

Q-covariance

To measure the dependence between two random variables X and Y one may use the covariance: cov(X,Y)

= E{XY)

-

E{X).E(Y)

provided that X, Y and X.Y are integrable functions. Hoeffding (1940) has extended this concept to a larger class of random variables by the formula: +oo

/>+oo

/ /

(F(x,y) -

Fiix^Fiiy^dxdy

-oo J—oo

valid when the above formula is finite. Krajka and Szynal [136] have shown that the concept of covariance may be further extended using the property of quantiles when one variable, for example Y is not integrable. Suppose that X is integrable. Let Y(p) be the quantile function of Y, and J be the indicator function. For X integrable and X and Y with continuous distributions functions, write : Lx,y{p)

=E((X-

E(X))I(Y

> y(p))).

152

Global Versus Local Dependence between Random

Variables

X-E(X)

Fig. 6.1

LXiY{p)

The function

LX,Y(P)-

= E(X - E(X)).I(Y

< y(p)).

Similarly one defines Ly,x{p) and Ly,x{p) for Y integrable and not necessarily integrable X.

The Q-covariance is defined by the change of variables (X, Y) i->- (X, p), where p is the quantile associated with Y : covQ(x,y)

= - / y(p)dLx,y{p) Jo

= - / Jo

y(p)dLx,y(p),

whenever one integral is finite. A more interpretable formula is given by :

c^, r i =

fi(x9(P„7™_-1>)

whenever Xy(P) ( . Y^\fV(P(P\ ~ 1) is integrable. Here P is a uniform variable on [0,1] independent of X, and /(.,.) is the density of the pairs (X, Y). When the two variables X and Y are integrable the three definitions coincide : cov{X,Y)

= covQ{X,Y)

=

covQ(Y,X).

Examples : Pairs of random variables (X,Y), density function m

where Y has a Cauchy

= 4TT?)' "°° < y < +0 °

153

Global Measures of Dependence

or also the density f(y) = ( y +"v+i > J/ > 0, r > 1, which has no moments up to order r — 1. For this latter distribution, the quantiles are easily obtained. Namely y(p) = -^-. Note that when r = 1, this distribution is obtained as the ratio of two exponential variables (Lachenbruch and Brogan (1971) [137]). 6.2.3

The coefficient

of linear correlation

p

Although we have discussed this coefficient in Chapter 2, at the risk of being redundant we shall briefly review this concept. This is needed to present further promising generalization. The coefficient p is formally defined as: p = corr{X,Y)

cov(X,Y)

=

\Jvar{X)var{Y) p is normalized, that is: -l
1.

The two limits (—1) and + 1 are attained in case when the dependence is maximal (negative or positive respectively). 6.2.3.1

The case when (X, Y) is bivariate normal

If the distribution of (X, Y) is a standardized bivariate normal (with mean 0 and variance equal to one for both X and Y), then the conditional density of Y given x is:

f(y/X = x)= , JKy/

'

1

exp (-%^Oi . P

V/2TT(1 - p2)

V

2(1

-?))

Therefore: E(Y/X

=

x)=px

and var(Y/X

= x) =

l-p2.

In this case p characterizes the distribution. We know also that in the family of the bivariate normal distribution, the condition p = 0 is equivalent to (X, Y) being independent. The equivalence

154

Global Versus Local Dependence between Random

Variables

is not valid if the distribution is not normal (except for the case of binary variables X and V). For other distributions, the connection between p,(x) = E(Y/X = x) and x may not be linear and the variance <J2(x) = var(Y/X = x) may not be constant. Hence p is not necessarily an adequate index to synthesize the relationship between X and Y. To characterize these connections for non-normal distributions, Bjerve and Doksum [21] [61] and Blyth [25] (cf Section 6.3.4) have defined a local linear correlation coefficient using the functions /3(x) = ^§~p- and a2(x). For certain distributions the index p may not even exist. For example, consider the bivariate Pareto distribution, described by Mardia [150] with the density function:

/(^) = ^ . ( f + f-D-a+2 (71 (72

"l

(6-1)

"2

where x > Q\ > 0 and y > 62 > 0 and a > 0. The marginal density function for X is: /i(x) = a.0?aT a + 1 , x > 6x > 0, a > 0.

(6.2)

It is easy to verify that : E(X) =a.0i—!—, a > 1 a—1

E{X2)=a.62—~,

a>2

«ar(X)=a.fl?(a_2)(o_1)a>a>2

caV(X,Y)=e1e2{a_2){a_ira>2 therefore p — \, a > 2 and does not exist when 0 < a < 2. The maximum correlation coefficient to be discussed next does not have this drawback.

155

Global Measures of Dependence

6.2.3.2

Correlation and extremal properties of normal distributions

The result proved by Klaassen and Wellner [131] states that for any bivariate normal distribution (X, Y), the maximum correlation coefficient, taken over all the transformations a and b from R to R : pM(X,Y)

=

suPa,bP{a{X),b{Y))

is equal to the absolute value of the linear correlation coefficient \p\. If a and b are restricted to monotone functions, Kimeldorf and Sampson [130] have proved that the maximum monotone correlation coefficient pm is again : pm(X,Y)

= \p\.

In fact Klaassen and Wellner show that the maximum pu is attained if and only if a and b are linear transformations of X and Y. This result extends straightforward to the normal copula model, that is the bivariate distributions (X, Y) with any marginal distributions G(x) and H{y) and such that ($-1(G(X)),$-1(H(Y))) has the standardized bivariate normal distribution with correlation coefficient p : PM(X,Y)=pM

6.2.3.3

( $ - 1 ( G ( X ) ) , $ - 1 ( i 7 ( Y ) ) ) = \p\.

p and the moment of inertia around the line D\ : {y — x}

In the family of the distributions which are bivariate normal with mean zero and variance equal to one, the upper Frechet bound F+(x,y) is the line Di : {y — x}. Indeed, in that case p = 1, therefore E(Y/X = x) = x, and var(Y/X = x) = 0, hence the distribution of F+(x,y) is concentrated on the line Di : {y = x}. If (X, Y) belongs to this family with correlation p, then (1 — p) is the expectation of the moment of inertia about the line {y = x}. Indeed, let (X, Y) be an arbitrary point on the plane then the moment of inertia d2 d2[(X,Y),D1]

=

(X-Y)2/2.

Therefore: E{d2) = 2{E{X2) - 2E{XY)

+ E{Y2))/2

= 1 - p.

156

Global Versus Local Dependence between Random

Variables

y tangent at x=0

Fig. 6.2

6.2.3.4

p is the slope of the tangent to the ellipse at x = 0.

A geometric

interpretation

Among numerous geometric representations of the correlation coefficients which actually go as far as Galton (1885), we shall cite the following recent one: consider two standardized variables (X*,Y*) ; Chatillon (1984) provides a rather wide class of bivariate distributions that have elliptically equi-density contours. Each ellipse is centered at the origin and the main axis of the ellipse coincides with the main diagonal of the plane of the domain of definition of these variables if p > 0, and the other diagonal when p < 0. Geometrically coefficient p is the slope of the tangent to the ellipse at the point x = 0. Note that due to the symmetry of the correlation coefficient, we can retrieve the same slope by appropriately interchanging X and Y indicated in Figure 6.2 which extends Figure 3 of Rodgers and Nickwander (1988).

157

Global Measures of Dependence

6.2.3.5

p and concepts of dependence

If the distribution of (X, Y) satisfies any of the concepts of positive dependence, for example if only the weak concept of Positive Quadrant Dependence (PQD) (cf Chapter 3) is imposed, then p will nevertheless be positive. Indeed in that case cov(X,Y) > 0 (Hoeffding lemma). If p is positive and (X, Y) is bivariate normal, then (X,Y) satisfies a stronger condition of Likelihood Ratio Dependence (LRD) [142]. 6.2.4

The ps of Spearman concept

and its connection

with the

PQD

This coefficient can be defined as the linear correlation coefficient between the two uniform variables U = Fi(X) and V = F2{Y), where F\ and F2 are the cumulative distribution function of respectively X and Y. Since U and V are uniform : E(U) = E(V) = 1/2 and var{U) = var(V) = 1/12. Therefore:

= 12.E{UV)-3 = 12 Jj0 112 uvdudv — 3 = 12(Jj 01]2 uvdudv - / [ 0 j l ] udu J[01] vdv).

(6.3)

ps is invariant to an increasing transformation of X and/or Y, it depends only on the ranks of the observations, ps/12 is an average measure on the PQD property on the whole plane [160]. Indeed, using (3.3), we know that if C(u, v) is the distribution function of the copula (U, V), then : cov(U, V) — /

(C(u,v) — uv)dudv.

•/[o,i] 2

Therefore : ps

=

12 /

(C(u,v) — uv)dudv

158

Global Versus Local Dependence between Random

Variables

= 12 J J2(F(x,y) - F1(x)F2(y))dF1.dF2.

(6.4)

The expression inside the parentheses corresponds to the evaluation of the PQD dependence at the point (x,y). ps is equal to 1 when F attains the upper Frechet bound, and to (—1) when it attains the lower Frechet bound. 6.2.4.1

A geometric interpretation of ps

Long and Krzysztofowicz [147] have shown that p$ is a measure of distance on the unit square which characterizes the density of the copula. Instead of considering only the squared distance d\ from a point (U, V) to the main diagonal D\ on the unit square as we did in the preceding section, they take : A = d\ - d\ where

dl = {U +

V-l)2/2

is the squared distance from (U, V) to the second diagonal D2 : v — 1 — u. We have A = {WV - 2{U + V) + l)/2. Hence: E(A) = 2E(UV) - 111 i.e. ps = 6£(A). 6.2.4.2

Estimation of ps

From Eq. (6.4), we can derive an estimator of ps given by (in obvious notation) :

Ps = 12 X)(F(iri) - F^).J^0))PiPj.

159

Global Measures of Dependence

6.2.5

Schweizer-Wolff's

index of dependence

An index closely related to Spearman's ps is the index GXY introduced by Schweizer and Wolff [197]. Instead of considering the difference C(u, v) — uv in formula 6.4, they use its absolute value to define : axY = 12 / 2 \C(u,v) — uv)\dudv J[o,\} OXY is a measure of the volume between the surfaces C(u, v) and u.v. Since Jj 0 112 |min(u,t;) — uv\dudv = ^ , we have the two equivalences : OXY — 0 ^ {X, Y) independent <JXY — 1 ^ X is a monotone function of Y. 6.2.6

The Kendall

r and its connection

with LRD

Two independent pairs of variables (X, Y) and (X',Y'), distribution F(.,.) are called concordant if

property with the same

(X-X'){Y-Y')>0 and discordant otherwise. The index r is denned as :

r

= =

P[(X-X'){Y-Y')>0]-P[{X-X')(Y-Y')<0] 2P[(X-X'){Y-Y')>0]-1.

(6.5)

Since P(X - X')(Y - Y') > 0) = P(X >X',Y > Y') + P(X <X',Y < Y') = 2P{X > X',Y >Y'). If the pair (X, Y) is fixed at (a;, y), so P(X'<x,Y'
= F(x,y)

If now, (X, Y) varies over the whole plane, we have : P(X' <X,Y'
160

Global Versus Local Dependence between Random

Variables

T = 4 f FdF-l.

(6.6)

In the particular case of Archimedean copulas with generator
r

=

4E{Z) - 1

1 - 4: /f \{z)d \{z)dz (6.7) Jo JO has the cumulative function K(z) = z — X(z), and =

where Z = F(X,Y)

Nelsen [160] proved that | represents an average measure of total positivity for the density / . He calculates : /iy2

+OO

/

j

pX2

/- + 00

l

j

-co J— oo J — oo J— co

[f(x2,y2)f(xi,yi)-f{x2,yi)f{xi,y2)}dxidyidx2dy2-

Let I

[f(x2,y2)f{xi,yi)

-

f{x2,yi)f{xi,y2)}dxidy1

-oo J —oo

then T(x2:y2)

= f{x2,y2)F(x2,y2)

-

D1F(x2,y2)D2F(x2,y2).

-

D1F(x,y)D2F(x,y)]dxdy.

Hence, + OO

/- + 00

/ /

[f(x,y)F{x,y)

-oo J — oo

Prom (6.6), the first term in the integral is | ( r + 1), and the second is | ( 1 - T ) . Therefore T = § .

6.2.6.1

Estimation of T

Given a sample of size n from the pair (X, Y), one can estimate r by :

f=

^(concordant pairs) — (((discordant pairs)

(I)

'

Global Measures of Dependence

161

If we want to test the independence between X and Y, we need to compute the expectation and the variance of f. Define the numerator K of f by : K = ^2 Zn< where: f 1 i£{X,Y)i and (X, Y)i> concordant { —i- otherwise. If the hypothesis of independence H0 is valid, then E(Zu') var(Zii.) = 1/2. Consequently E(K) = 0 and var{K) = E{K2) =

E ( S i = i Z)"=j+i zw) lations, one finds :

= 0 and

• After some tedious but straightforward calcu-

= v

'

n(n-l)(2n + 5) 18

hence var{r) =

var(K) n(n - l ) / 2

2(2n + 5) 9n(n-l)'

For n > 10 the distribution of f is normal with a sufficient approximation and consequently one can presumably test the hypothesis of independence using the standard procedure. See, however, the warning by Hallin and Seoh (1998) [91]. 6.2.7

The Blomqvist

medial

coefficient

This coefficient, also known as quadrant test of Blomqvist, evaluates the dependence at the "center" of a distribution. If X and Y are independent, then, in particular, F ( l / 2 , 1 / 2 ) = F1(l/2)F2(l/2) = 1/4 The coefficient of Blomqvist is then defined as : /3 = 4 F ( l / 2 , l / 2 ) - l . 6.2.8

r, ps, /3 and ordering

on the

distributions

Lehmann (1963) [142] and later Tchen (1980) [216] have proved that Kendall's T, Spearman's ps, and Blomqvist's /? are monotone functions of the underlying bivariate distributions F. If two bivariate distributions G and H have the same marginals, and if G(x,y) < H{x,y) for all x and y, then the values of the three coefficients are smaller for G than for H.

162

Global Versus Local Dependence between Random

6.2.9

Constructing

other global

Variables

measures

We note that there exists a multitude of measures in the literature related to the linear correlation coefficient p, Spearman's ps and Kendall's r which we have already discussed. Many non-parametric measures of dependence are based on the distance between a copula to the independent copula or to maximal dependent copula. For example the measure of the type : F(a,y)-Fi(a;)Ja(y) min(F1(x),F2(y)) - F1(x)F2(y) with or without normalization, with or without various averaging over the range of the distribution (either the joint or the product of the marginals). These are given by Schweiwer and Wolff (1981) [197], Tuncer and Sungur (1980) [220], Reimann (1987) [178], and earlier by Hoeffding (1948) [97]. Properties of these measures are closely related to Spearman's ps and Kendall's r. 6.2.10

Indices for more than two

variables

Wolff [225] has extended the indices axY and ps to measure the dependence within the vector X = {X±,...,Xn) (n > 2) with the distribution F. Let C be the copula associated to F. On the unit n-cube In = [0,1]", let us consider the two copulas corresponding to the independence and to the maximal dependence (the upper Frechet bound):

C°(ui,...,un)=n?=1«i and C + (ui,...,u„) =

Min(ui,...,un)

and the function corresponding to the lower Frechet bound: C~(ui,...,un)

= Max(ui + ... + u„ - n + 1,0).

Let an and bn be the volumes : an=

(Min(u1,...,un)-U^=1Ui)dui...dun

163

Global Measures of Dependence

and bn=

Ji„

(II"=1Uj - Max(ui + ... + un -n +

l,0))dui...dun.

Then the indices an and pns are defined as: an = a~1 /

\C(ui,...,un)

-C°(ui,...,u„)\dui...dun

and PnS = an1 /

(C(ui,...,u„)-C°(ui,...,un))dui...du„.

The index an satisfies all the axioms of Renyi (adapted to the case when n > 2). The index pns can be negative and does not satisfy the axiom on independence. However, if n is large the functions C~ and C° are not "far" and : < PnS < 1

an where the quantity — £*• is 0(^-) and tends to zero when n increases. In the same way Kendall's r can be extended to : T

"

=

O"- 1 - 1 V 2 " /

( < - ? »( Ul > U2 >"-> u n) dC '«( Ul > U2 '"•>"") ~

1

)

(Jouini and Clemen [120]). Again, one sees that the lower value of r n = ~ 2 n - i - i t e n c ^ s rapidly to zero when n increases. 6.2.11 6.2.11.1

Mutual information, measures

relative entropy

and

derivatives

Definitions

(1) If X is a random variable, with density fi(x), the measure of uncertainty is defined by : Ex = -

then the entropy or

fi(x)logfi(x)dx.

If we consider all the distributions with the density / defined on a compact set A of R, i.e. fA fdx = 1 and / = 0 on R — A, then the entropy is maximal when / is the uniform density on A [109].

164

Global Versus Local Dependence between Random

Variables

(2) If (X, Y) is a pair of random variables with the density f(x, y), and the marginal densities fi(x) and f2(y), then the entropy for this pair is : Ex,Y = ~

f{x,y)\og(f{x,y))dxdy.

This entropy is maximum, when X and Y are independent. This definition can be generalized with an n-vector (Xi,..., Xn) in place

of(X,Y). (3) One can show [134] that - / / f(x,y)\og(f{x,y))dxdy

<~

f(x,y)log{f1{x).f2(y))dx.dy.

The mutual information [20] or relative entropy [109], [110], [121] is then defined as the difference between these two quantities :

'*•'= f f'M^JW***-

(6 8)

'

If the components of (X,Y) are independent, then 5X,Y is zero, and conversely when the dependence is maximal, SX,Y approaches infinity. To normalize this index, Joe [110] defines : (J* = ( l - e x p ( - 2 5 ) ) * .

(6.9)

The index 5* is confined to the interval [0,1], and in the case when the pair (X, Y) is bivariate normal is equal to the absolute value of the linear correlation coefficient |p|. Indeed, given (x,y)'~7V(Ml)At2)S) where S =

a{

paia2

/9<7i<72

o\

we have 5 = - | . l o g - \ \ and consequently S* = \p\.

165

Global Measures of Dependence

6.2.11.2

Examples

(1) Generally, if X is a p-random vector of regressors and Y is the vector of the dependent variable, so that (y,X)'~JV(My,/iX,E) with S =

'YX

Sxy

^xx

then 8yX =

WX^XX^XY

1/2

(2) If (X, Y) is a p + g-random normal vector with the covariance matrix: S =

^xx Syx

Syy

_M_ then, 8 = — l-logj^—'TT^—rHowever one can expand (see e.g. Searle [199] page 43) the determinant of this block-partitioned matrix as follows : IS I

= =

'YY\ Hxx—SvySvvS -'XX XY^YY^YXl XX^XY^YY^YX I / S ^XX

ISYY\

Hence 5 = --Aog\I

-

S^SxySyySyxl

If A* is a strictly positive eigenvalue of S ^ S x y S y y S y x (it is also less than 1), then 1 — Aj is a non-zero eigenvalue of I — S ^ S x y S y y S y x and therefore its determinant is : n?=1(i-A0. Hence [i-n?=1(i-Ai)]1/2 where s is the number of)f non-zero ei eigenvalues. In fact A; = pf is a canonical correlation coefficient (see e.g., Johnson and Wichern

166

Global Versus Local Dependence between Random

Variables

[117], and the preceding formula indicates that the normalization of the relative entropy depends only on the non-zero eigenvalues {Ai,..., A s }. Lin [145] has shown that it is a property of functions which are invariant with respect to a linear transformation (location and scale changes) of X and Y. (3) Bivariate Pareto distribution We can calculate the entropy for the pair (X, Y). For X, we have : /»OC

Ex

=

- /

/i(a;)log/i(z)
/»oo

=

-

=

-log(a) - a l o g 0 i + (a + 1) /

-

- l o g a - a l o g < ? i + (a + l ) [ - + log#i] a logl?i - l o g a + l + - . a

=

fi{x)[loga +alogOi - (a +

l).log(x)]dx fi(x)dx

(6.10)

Similarly: EX,Y = -[loga + log(a + 1) + l o g ^ + log0 2 + (a + 2)[- + a o+l Hence: 6X,Y

= log ^

a

- (a + 2)[i + - L - ] + 2(1 + J). a

a +1

a

Therefore 5X,Y is defined for a > 0, over a larger domain than p. (4) Bivariate Inverted Dirichlet Distribution Tiao and Guttman [218] have constructed a multivariate inverted Dirichlet distribution, whose density for the case of two variables is given by : r(m1+m2+m3) 1 [X

'

V)

a,mi-i y ™ a -i

~ r ( m 1 ) r ( m 2 ) r ( m 3 ) • (1 + X + y)mi+rn2+m> '

X

~

U

'

V

~

U

'

(6.11) Kapur and Dhande [121] have shown that if mi = mi = 1 and mz = a, and if one transforms U = 9i(X + 1) and V = ^ ( ^ + 1)> then the pair (U, V) has a bivariate Pareto distribution when (X, Y)

Global Measures of Dependence

167

has a bivariate inverted Dirichlet distribution. The coefficient of correlation is : /

(mim2)1/2 (m3+m1-l)1/2(m3+m2-1)1/2

^

P[X,y)

and the relative entropy is given by [121] : r *'y

, 0gl

,T(mi + m 2 + m 3 ) r 3 (m 3 ) r(mi+m3)r(m2+m3) J r'(mj + m 3 ) r(mj + m 3 )

f-^

. .r'(mi+m2+m3) T'{m3) - ( m i + m2 + m 3 ) — • - mz— {-. r(m1+m2+m3) i(m3) (6.12) In the special case when mi = m 2 = m and m 3 = a, we have : p(x,y) =

m m +a— 1

and .

5x Y = lQ

<

g

T(2m + a ) r ( a ) . i + rl^(m 2 / m x+ a)

/•o^ , „\r'(2m+a) - ( 2 m + Q) r(2m+a) -

6.2.11.3

n.

2 m + a

T'{m + a)

W1 (ma.+^a)

„r'(a) T(a) •

a

Lm's measure of association

Since the measures which are invariant with respect to a linear transformation depend only on the {A* i = l , . . . , s } , the non-zero eigenvalues of S ^ x S x y S y y S y x , Lin [145] proposed to construct a measure of association between X and Y as a function of these eigenvalues. Definition : Let / be a differentiable and strictly increasing function, / : [0,1] H- [0,1] with /(0) = 0 and / ( l ) = 1 and let g be a differentiable and monotone function g : [0,1] i-> [a, b], with 0 < a < b < oo and either : - g(0) = a and g(l) = b - or
pf,g(X,Y)=g-1\J2ci9[f(Xi))

168

Global Versus Local Dependence between Random

Variables

if s > 0, and otherwise Pf,g(X, Y) = 0, with the constraints Q > 0 and E"=i °i = L But if a = 0 and 6 = oo, the condition ^ i = i c* = 1 is not required; however at least one of the a, i = 1,..., s, must be positive. Any pft9 defined as above possesses the following properties. (1) Symmetry : pf>g = pgJ. (2) 0 < pfi9 < 1. Indeed, the eigenvalues A* are between zero and one. (3) pft9 = 0 O S x y = 0, therefore in the case where X and Y are normal vectors this is equivalent to (X,Y) being independent. (4) If Y = HX where H is a p x q matrix of rank p, then pfi9 = 1. Example : The measure 5*x Y defined by Joe (Eq. 6.9) is a pfg measure with f = I and g(y) = - l o g ( l - y), and c; = 1 for all i. In that case 5*XY = 1 implies that there exists a matrix H of rank p such that Y = HX. 6.2.11.4

Zografos's measure of association

Starting from the Lin's measure of dependence, Zografos [233] proposed a new measure of dependence r]x,Y between the two random vectors X and Y equal to pft9(Sx,Sy), where Sx and Sy are the score function vectors: s

_ dlogf(x,y) dx

s

=

dlogf(x,y) dy

_ (dlogf(x,y) V

dx

i

dlogf(x,y)

_ fdlogf{x,y) V

dyi

dx

'""'

p

j ip

dlogf(x,y)\' '

'

d

Vq

1?

Let f{x,y) satisfy certain regularity conditions established by Papathanassiou [164] namely : (1) continuous partial derivatives: dlog f(x,y)

w • _ -,

aiog/(s,y) w • _ i

_ d2\ogf(x,y)

w•

, .

are continuous. (2) vanishing on the boundaries of the domain of definition: / is defined on a set E, which is a union of open connected sets in Rp+q , and f(x, y) tends to zero monotonically as x or y approaches the boundary of E along the coordinates axis. (3) The Fisher's type positive semi-definite block symmetric information matrix is non-singular :

169

Global Measures of Dependence

// =

^11

Il2

hi

I22

where dlogf{x,y) dxi

hi=E

I22 =

I\2 = E

E

dlogf{x,y) dxi

i, j = l,...,p,

dlogf(x,y)dlogf(x,y) dyi dyj

dlog f(x,y) dxi

dlog

i, j = l,...,q

f{x,y)

= 1,-,P, 3 = I , - , ? -

dyj

In that case E(Sx) = E(Sy) = 0 and In = var(Sx), I22 = var(Sy) and J12 = coi^Sx.Sy). Since Sx and <% play the role of X and Y in the definition of pf9, the matrix E ^ E x y E y y E y x becomes I^^I^hi and the A; are the eigenvalues of the latter matrix. The measure r)x,Y has the same properties as pfi9 : - symmetry " 0 < T]X,Y < 1

- VX,Y — 0 <£> (X,Y) Section 6.2.11.3 :

are independent.

VX,Y

Indeed using property 3 of

= 0 & I12 = 0.

This is, however, equivalent to ,dlogf(x,y) (

dXi

dlogf(x,y)^ •

dVj

. ) =

0 V

' .

. V

' -

Also since the differentiation under the sign integral is justified, this is equivalent to :

II

d2 log f(x,y) -f(x,y)dxdy dxidyj

= 0,Vi ,V j

and hence —dx dv- = ® a'e' m x an<^ ^' ^ n e ^ a s t r e l a t i ° n 1S t n u s equivalent to the independence of X and Y. - The measure is invariant under a linear transformation.

170

Global Versus Local Dependence between Random

Variables

-lip = q and Y = AX where A is a non-singular matrix, then (see property 4 in Section 6.2.11.3). Examples :

TJX,Y

= 0

(1) Let (X, Y) be a bivariate random vector: Then If becomes : It = S - 1 =

1

1

°5

rfi'

1

The eigenvalue of I^I^I^hi is p2- With g = Id , and f(x) y/x, one obtains TJX,Y = |p|, which is S*, the measure of Joe. (2) (X, Y) is a multivariate random vector: In that case, using a theorem of Mayer-Wolf [153], we obtain : J/ = E - l and from Anderson [6] we have: In I12I22 I21 — S x y S y y S y X E ^ x . These two products of matrices have therefore the same eigenvalues A; . If we take / = Id and g(y) = — log(l - y), and c; = 1 then : r?x,y = l - n | = 1 ( l - A i ) i-e r]x,Y is the square of the Joe measure 5X Y • (3) The bivariate inverted Dirichlet distribution : In the particular case when mi = rn,2 = m and m^ = o, Zografos [233] has shown that : a(a+l) If = 2m + a + 1"

m+a+3 m-2

-1

-1 m+a+3 m-2

With / = g = Id, a = 0 and c2 = 1, VX,Y = ( m +~+ 3 ) 2 which is a simpler measure that the relative entropy suggested by Kapur and Dhande [121] .

Local Indices of Dependence

6.3

171

Local Indices of Dependence

6.3.1

Motivation

We have seen that ps is an average measure of the PQD dependence. . The following example presented in Kotz et al. [133] shows that a distribution with a high ps may not be PQD. Y/X 1 2 3

1 0.25 0.01 0.02 0.28

2 0.10 0.03 0.36 0.39

3 0.01 0.22 0.10 0.33

0.36 0.26 0.38

Here one can estimate ps by : ps = 12 ^

( ^ T J )

-

F$i).FW))PiPj

we obtain : As = 0.53.

However the distribution is not PQD. Indeed: F(2,2) = 0.39 is less than Fi(2)F 2 (2) = 0.4154. 6.3.2

Local definition

of the

dependence

Our aim is to define local dependence and "remaining" dependence. (We use this last term to indicate dependence on a part of R2, specifically in case of survival variables in an orthant beyond a certain point. If V(xo,yo) is an open neighborhood of (xo,yo), then a distribution F(x,y) is PQD in the neighborhood V(x0,y0) provided V(x,y) 6 V(x0,yo) we have S{x, y) >

S(x)S(y).

If V{x0,yo)

=}x0,+oo[x]y0,+oo[

we then arrive at the remaining PQD. In the same manner a local or remaining LRD can be defined.

172

Global Versus Local Dependence between Random

Variables

Example : the Cauchy distribution is PQD in the first and third quadrant and NQD (negative quadrant dependence) in the others. 6.3.3

Local ps and r

We can restrict ps and r to an open neighborhood of (£0,2/0), and define:

PS,(x0,„o)-

_

U

IIv{X0,yo)(C(U'V) JJ

~

UV dudv

)

dudv

and T

(xo,yo) —

rr J

Ar,

JV(xo,yo)aL/

In the case when V(x0,y0) =]x0,+oo]x]y0,+oo[, it is straightforward to estimate T(x0,yo) DV c o u n t i n g the remaining concordant and discordant pairs, and to estimate the variance of this estimator from no, the number of remaining observations. 6.3.4

Local correlation

coefficient

of Bjerve

6.3.4.1

Motivation and historical remarks

and

Doksum

Since K.Pearson's paper (1905), many authors described situations where the linear correlation coefficient takes unsuccessfully into account the association between two random variables X and Y . The reason for this is that either the dependence does not exist on the whole plane or it is not linear, or the variance of Y given X = x is not constant (heteroscedasticity). Blyth [25] reproduces in his paper an example extracted from "On the general theory of skew correlation and non-linear regression" (1905), where K. Pearson analyses a data set, consisting of measurements of auricular heights of 2272 schools girls according to their ages (between 2 and 23). In this example, Pearson proposes to fit the data by a cubic regression curve : (JL*{X)=

0.28 + 0.723.x* - 0,0296.zJ - 0,00222a^

where x is the age and y the auricular height, * indicate that the data are centered around the empirical mean, and fi(x) is E(Y/X — x). Pearson also plots the empirical conditional standard deviations, scaled by aY, that

173

Local Indices of Dependence

Blyth reconstructs them to be : &{x) - 4.5 + 0.17a; where a2(x) = var(Y/X 6.3.4.2

= x).

Definition, properties and limits

(1) Definition: If (X,Y) is bivariate normal regression curve is simply E{Y/X

N{IIX,HY,O-X,0Y)

the

= x) = a + l3.x + e,

with var(e) = a2. The linear correlation coefficient is then :
(o-xp)2+af

If (X, Y) is non-normal, a local extension of p is defined by : PKX)

{axp{x)Y+a{xY

where (3(x) = g£', (assuming differentiability of p.(x)). (2) p(x) is standardized, that is - 1 < p{x) < 1. (3) The components of (X, Y) being independent implies that p(x) = OWx. (4) p(x) ± 1 for almost all x is equivalent to Y being a function of X. (5) In general p(x) is not symmetric, but it is possible to construct a symmetrized version p~ by p(x) — sign(p(x))y/p(x).p(y) if p(x) and p(y) have the same sign and p(x) = 0 otherwise. (6) p(x) is scale-free but not marginal-free, that is linear transformations of X and Y, X* = aX + b and Y* — cY + d, with c and d having the same sign, leaves p(x) unchanged, but the transformations U = Fi(X) and V = F2(Y) result in p(u) which is different from p(x). (7) the PRD (positive regression dependence) and p{x) : If p(x) > 0 t n e n PRD is valid. We can also define a local PRD when p(x) is positive in a neighborhood of (XQ, yo).

174

6.3.4.3

Global Versus Local Dependence between Random

Variables

Estimations and properties of the estimators

Bjerve and Doksum [21] use a non-parametric estimation method for the parameters /x(x) = E(Y/X = x), (3{x) = ^ - and a{x) = y/var{Y/X = x). Let K(u) be the Epanechnikov kernel, K(u) = 0.75(1 — M 2 )li u t< 1 (u), and XQ be a point of a subdivision on the x-axis. y = a(x0) + b(x0)x is a weighted least square line computed from the sample {(xt,Yi) i = 1, ...,n}, with the weight Wi = K((xi - x0)/h), where h = sx is the empirical standard deviation of {xi, i = l , . . . , n } . The estimates are therefore : fx(x0) = a(x0) + b(x0)x0 and

Kxo) - b(xo)Similarly to estimate a2(XQ), consider the sample {(xi,e2), i = 1,..., n)}, where e.i = Yi — p.(xi) is the i-th residual, and fit the line y = c(xo) + d(xo)x with the same weights u>i as above. The estimate of O~2(XQ) is then C(XQ) + d(xo)xo and the estimate of the local correlation coefficient is -, s p{xo) =

sxP(xo) , ^s2xp2(x0)+&2(x0)

The above procedure is then repeated for all the points of the subdivision. The delta-method allows us to estimate the standard deviation of p(x). One expands vV + t around the values so = o~xP{xi) a n d to = o-2(xi), and uses the fact that /3(xi) and a{xi) are independent. In a subsequent paper Doksum et al. [61] propose to use two other non-parametric methods to estimate p(x):the nearest-neighbor method method based on the Gasser-Miiller [79] kernel. They provide conditions for consistency and asymptotic normality of p(x). They choose an optimal bandwidth by using a parametric model and searching for the optimal bandwidth under this model.

175

Local Indices of Dependence

6.3.5

Correlation

ratio

Renyi (1959) [179] motivated by Kolmogorov defines the correlation ratio : KY(X)

=

Var(E(Y/X)) Var(Y)

When the distribution of (X, Y) is standardized bivariate normal, Kx(Y) = p. One can show that Ky{X) = supg\p(Y,g(X))\ for all measurable functions g, such that g(X) exists. One also deduces that : KY(X) 6.3.6

= \P(Y,g(X))\

Local dependence

ittg(X)

function

= aE(Y/X)

+b

of Bairamov

and

(a.e). Kotz

Bairamov et al. [13] have defined a "local dependence function" H(x,y), which provides a local point of view on dependence at a point (x,y). If H(y) = E(X/Y = y) and p,(x) = E(Y/X = x), then : H{x,y)=

WX - WW ^E{X-n{y)y.E{Y-n{x)Y

*x))

H(x, y) is obtained from the expression of the linear correlation coefficient by replacing expectations E(X) and E(Y) by the conditional expectations n(y) = E(X/Y = y) and p,(x) = E(Y/X = x) respectively. After some algebraic manipulations the expression can be rewritten as: H(x,y)= where * * ( „ ) = ^ g >

, ^(l and

P+fxMvrW + ipx(y))(l

VY(X)

=

(6.13) + V>Y(x))

^ M .

For a further analysis of Equation (6.13), let a < ipx(y) < b and a <
~

V ( l + i 2 )(l + * 2 )

The partial derivative of h(t, z) w.r.t. z is : dh _

Tz'

t — zp

(l + z a j t ^ + P

176

Global Versus Local Dependence between Random

Variables

Fig. 6.3 Graph of H(x,y) corresponding to standardized normal distributions with p = 0.95, p = 0.5, p = 0.005.

and the other partial derivative is obtained by exchanging z and t. These partial derivatives are zero at the point t = 0 and z = 0. Calculating the second derivatives at the point (0,0) we find : (82hd2h

. dh .2\.n

n.

2

1

which is negative if \p\ < 1. Therefore h(t, z) has a saddle point at the point (0,0), that is H{x,y) has a saddle point at the point (x*,y*) satisfying: ifx{y*) = 0, i.e. E(X) - »{y*) and,
177

Local Indices of Dependence

6.3.7

Measures

of the tail

dependence

Coles et al. [50] have proposed two indices to measure tail dependence (see Chapter 3, Section 3.2.10) and a diagnosis of such a dependence. Recall that for a copula C(u,v), the upper tail dependence is defined as : C(u,u) d = limu_n_1- u This formula can be rewritten as : S

=

2

„

1 -2u + C(u,u) 1-u 1-C(u,u) — Zim u _j.i1-u

limu-)-l-

2-Km^1-tog1g("'u). logu

(6.14)

Coles et al. check the behaviour of the function 6(u) = 2 —° S i 0 g„ to diagnose tail dependence. ( 5 — Zim„_>i_<5(u).) For independent variables S(u) — 0, for the Frechet lower bound C~ (u, u) max(0,2u — 1), S(u) = 1, and for Gumbel-Hougaard family 5(u) = 2 - 2a. More generally S(u) is constant (different from zero) for bivariate extreme value distributions. For asymptotically independent variables (for example for the normal copula model) S(u) may be a complicated function of u, which tends to zero when u tends to one, and tends very slowly (and ultimately abruptly) if the correlation between U and V is high. So an estimation of 6(u) may be a bad indicator of tail dependence since the empirical observations are taken when u < 1. To complete the diagnosis, Cole et al. propose to use another function: 2/o 5 (l - u) S(u) = — ^ b x logC(u,u)

. -

which is in the range of [—1, +1]. For asymptotically dependent variables the limit 5(u) when u tends to one is S = 1. For independent variables 5(u) = 0 and therefore 5 = 0. For asymptotically independent variables S measures the strength of dependence, for example for the normal copula model S is equal to p, the correlation coefficient. The two indices (5, S) can therefore be used to identify extremal dependence: (S > 0, S = 1) signifies asymptotic dependence in which case the

178

Global Versus Local Dependence between Random

Variables

value of 8 determines a measure of tail dependence; (5 = 0,5 < 1) signifies asymptotic independence, in which case the value of 6 determines the strength of dependence. 6.3.8

Several local indices applicable

in survival

analysis

In the field of survival analysis, there is a need for time-dependent measures of dependence, to identify, for example in medical studies, the time of maximal association between the interval from remission to relapse and the next interval from relapse to death, or to determine the genetic character of a disease by comparing the degree of association between the lifetimes of monozygotic or dizygotic twins [100]. 6.3.8.1

The covariance function of Prentice and Cai

With the aim to propose an estimator of the bivariate survival function in case of censored data, Prentice and Cai [173] have denned a covariance function characterizing the dependence between two failure time variables independent of the marginal distributions. They were searching for measures that express the dependence between (X, Y) as a function of follow-up durations x and y. Let Ni(x) and N2(y) be the couting processes (Ni(x) = 1 if X < x and Ni(x) = 0 otherwise), H0i{x) and H02(y) be the cumulative marginal hazards and Mi (a;) = Ni(x) - H0i(x A X) and M2(y) = N2(y) — #02(2/ A Y), the martingales arising in the decomposition of Ni, i = l,2, with respect to the filtration defined by X (or Y), then the covariance function is defined as : C(x,y) = cov(M1(x),M2(y))

=

E(M1{x).M2{y)).

C(x,y) and the marginals allow us to retrieve the survival function. Prentice and Cai show that : S(x,y)

= S1(x)S2(y){l+

f

fy{S1(x)S2{y))-1C(du,dv)}.

Jo Jo

For example with the exponential marginals Si(x) = exp(-x) and S2(y) = exp(-y) and with C(du,dv) — 9exp(-2(u + v))dudv, one obtains: S{x, y) = exp{-(x

+ J / ) ) { 1 + 0.(1 - ea:p(-i))(l -

exp(-y))}

which is a bivariate exponential distribution considered by Gumbel [88].

179

Local Indices of Dependence

Shih and Louis [206] have considered the estimator of this covariance function in case of frailty models to search for an early and late dependence for censored survival data on AIDS. 6.3.8.2

The conditional covariance rate of Dabrowska

Dabrowska et al. [53], in the case where (X,Y) are two survival variables, proposed to use "the conditional covariance rate" : l(x,y) = D1D2Log(S)

= h(x,y) -

hi(x,y)h2(x,y)

or equivalent ly : l(x,y) = -Dih2{x,y) where hi(x,y)

and h2(x,y)

=

-D2hi(x,y)

are two conditional hazards (see Chapter 1).

(1) : Interpretation from the frailty model (Chapter 4, section 4.5.5): Starting from Equation (4.21) : S(x,y) = / / exp(-w1Hw(x)

+H2o(y))dG(wi,w2)

we obtain for the partial derivatives: DiS = - / / Wihio(x)exp(-wiHw(x)+H20(y))dG(wi,w2),

i = 1,2,

and for the conditional hazards : h(

DiS

\-

m(x,y)-

- h ( \$ J wiexp(-w1Hio(x)

-nl0[x)

s =

,

,

J

JWJP(X

M[X)

= hi0(x)E(Wi/X

jjexp{_WiHio{x)

+ +

H20(y))dG(w1,w2) H2o{y))dG{wuW2)

> x,Y > y)/wi,w2)dG(wi,w2) JfP(X>x,Y>y)/wl,w2)dG(w1,w2) >x,Y>y),i=l,2. (6.15)

Using the same procedure, we find for the bivariate hazard: h(x,y) = h10(x)h20(y)E(W1W2/X

> x,Y > y).

Therefore : l{x,y) = hw{x)h20{y)cov(W1,W2/X

> x,Y > y).

180

Global Versus Local Dependence between Random

Variables

An analogous proof is provided in Anderson et al. [6], but only for one frailty variable. (2) Interpretation : A discretized version of I: Suppose that x takes on the values 1,2,..., i, i + 1,..., k and y the values 1,2,...,j,j + l,...,m. Denote Sij = P(x >i,Y> j) and define CO Wij

— Si,jSj+l,j+l — c c ' >->i+l,jdi,j+l

i.e. SSij is an odds-ratio among the four adjacent cells, which measures the interaction among these cells. Define now LLij = log(SSij). LL^ is then a discretized version of l{x,y). Indeed: H

Log(S

dx-+0,dy-yO

S

+dy\^

\Jx+dx,y>Jx,y+dy

DiD2Log{s)

/

(3) I and the joint distribution : - The function l(x,y) is symmetric. - / and the two marginals completely determine the distribution. Indeed, S(x,y) = Si(x)S2(y)e*P

/

l(u,v)dudv

Jo Jo

-1 is not marginal-free : Indeed, if C represents the copula function associated with the given survival functions, and c is the Copula's density : S(x,y) =

C(S1(x),S2(y))

then the density function is : f(x,y)

^ D1D2S(x,y)

=

c(S1(x),S2(y))fi(x)f2(y)

and the bivariate and conditional hazards are : h(x,y)

f(x,y) S(x,y)

DiS hi(x,y) = - — =

c(S1(x),S2(y))f1(x)f2(y) C(Sx,S2 DiC{SuS2)fi

V . = l,2.

181

Local Indices of Dependence

Thus ,

.

. .

fffc

DxCD2C

1 = h - h1h2 = 7 1 / 2 ( 7 ;

-Qi—)

i.e. I = 7*102*0

where lc represents the conditional covariance rate for the copula. (4) I and positive dependence: - If X and Y are independent, this implies at I = 0 and vice-versa. - The condition l{x,y) > 0,V(x,y) , corresponds to positive dependence. More precisely if l(x,y) > 0, then h2(x,y) is a decreasing function of x, that is equivalent to the fact that S(x,y) is TP2 (DTP(l.l) dependence, see Chapter 3, Section 3.2.7). (5) / and the upper Frechet bound : The density f+(u, v) of the upper Frechet bound on the unit square can be viewed as the Dirac measure on the first diagonal Su(v) (see discussion of Frechet bounds in Chapter 4). What is the limit of / ? If this limit exists, it is : 1 [u v)

' ~

s+

s+.s+

(the symbol + corresponds to the upper Frechet bound). We have 5=1+^-^1-^2. Therefore : DXS = DiF - Di-P1!. KeieDiF+(u,v) = l[ 0)0 ](u), hence DiS+(u,v) = l[ 0iU ](u)-l[ 0)1 ](u) + i.e. JDi5 (u,u) = - l ] „ | l t ( u ) . In the same manner one can deduce that D2S+(u,v) = — l]u,i[{v)Therefore D1S(u,v)D2S(u,v) = 0 and l(u,v) = ^ - . The upper bound of I is thus concentrated on the diagonal of the unit square.

(6) A multivariate generalization [52] :

182

Global Versus Local Dependence between Random

Fig. 6.4

Variables

/ for Clayton's copula with high dependence, a = 200.

Let S be the survival function of a non-negative random vector X = (XuX2,...,Xn). Then: 5 = 5i52,...,Mi 2 Afi3...Mi23-Afi2.. n , where Si, i = 1,..., n are the univariate marginal survival functions of S and the interactions terms Mij are defined from the bivariate marginals Sij = SiSjMif, or also Mtj = exp f[0tX.]x[0tXj]dLogSij, the last term being Mit2,...,n = exp L , dLogS. The functions DiDjLogS,...,dLogS are then multivariate generalizations of the conditional covariance rate l(x, y). The increments of LogS on the hyper-cube [0, x] can be interpreted in different ways. For example: ( •/[0.x]

dLogS = E | a > 2 | ( - i r - a [LogSa(xa)

-

Log(UieaSi(xi))]

Local Indices of Dependence

183

where a denotes a set of indices: 0 ^ a C v = {1,2, ...,n}, and x a denotes the coordinates of the point x whose indices fall into the set a. The right-hand side can be interpreted as a weighted sum of dependence functions comparing each joint marginal Sa with its one-dimendional marginals. Another form is : f

dLogS = - S ( r i i . . . , T l ) ( - l ) ' - 1 ( / - l ) ! [LogS(x) -

Log(Illi=1STi(xTi)}

where the sums is over the partitions r = ( T I , . . . , T ( ) , I > 1, of the set v = {1,2, ...,n} into non-empty mutually disjoint blocks. The right-hand side can be interpreted as a weighted sum of measures of dependence in which the term associated with the partition r is a measure of deviation of the survival function S from SH0 = STl.ST2...STl. This measure takes into account the dependence which is not contained in the lower dimensional marginals. In [52], Dabrowska studies the properties of these dependence functions and proposes a class of tests of independence for censored data. 6.3.8.3

0 : the ratio of two conditional hazards

The index 0 is the ratio of two conditional hazards : Q x

( >y)=ur—r h2(x,y)

i.e., using (1.1)

°^V)

_ D2D1S(x,y)S(x,y) ~ DlS(x,y)D2S(x,y) •

(6-16)

< 6 - 17 )

Alternatively

hi(x,y)h2(x,y) This index was originally defined and used by Clayton and Cuzick [46], Oakes [163] and Anderson et al. [6].

184

Global Versus Local Dependence between Random

Variables

(1) Interpretation : © was defined by Clayton and Cuzick [46] as an odds-ratio. Suppose we are dealing with two discrete failure time variables taking on the values i = 1,2,..., n and j — 1,2,..., m. Denote by Py, the relative frequency of a failure of the two components in the cell (i,j), then the discrete analogue of 0 is: £

e ( M ) ^ f i

y ^ p '

f c

.

(6.19)

(2) 6 and the joint distribution : - 0 is symmetric, - 0 is marginal-free. Indeed, using the formulas shown in the preceding section, DiD2S = c(SuS2)hh and DiS = £>iC(Si,S 2 )/i and similarly for D2S, we obtain : 0 = %DC£)1Q where C is the survival copula associated with S. - 0 and the two conditional hazards hi and h2 determine the joint distribution : Indeed, using equations (6.18) and (1.2), we arrive at l

0-1=

hih2

We know that I together with Si and S2 determines S. S

SiS2 However using hi = -^^-,

= [ [ l(u,v)dudv. Jo Jo we obtain :

S — = exp(<JI

(6.20)

fx hi(u,y)du).

(6.21)

Jo

Thus the two preceding equations allow us to determine S2. In the same manner, Si can be determined. (3) 0 and dependence : - If (X, Y) are independent, 0 = 1. Namely, in that case /i 2 /i = h2 = hy, the marginal hazard of Y. The converse is also true. The stronger the dependence between X and Y is, the higher is the value of 0 . - 0 and the upper Frechet bound :

185

Local Indices of Dependence

Using the results presented in the preceding section dealing with Frechet bound of I, we observe that 0 approaches infinity when S tends to its upper Frechet bound. - The condition 0 > 1 is equivalent to S being TP2: indeed, in this case I > 0 and therefore , using Eq.(1.3), — -D1/12 > 0, i.e. h2 is a decreasing function of z, this is equivalent to S being T P 2 (Chapter 3, Section 3.2.7 ). -Oakes has shown that g^y is a conditional version of Kendall's r. More precisely, using the same notations that in Section 6.2.6, ®{,X'y\Z] = Q(x,y) + 1

EiZivlminiX^Xi^miniYuYi.)).

Therefore 0 also measures a "remaining" dependence, but this is not exactly the same as the measure obtained with r(xo,yo) in Section 6.4.3. - 0 being constant is equivalent to the condition

S(x,y)i-& = S±-& +

Sl-&-l.

Proof : One can write Equation (6.17) in the form : D1D2S £>iS

=r)D2S

S

'

Integrating with respect to y, we obtain : logi-DtS)

= QLogS + k(x) .

We shall evaluate the function k(x), at the point (x,0): l o g t - D i S i ) = QLogiSi) + k(x) . Substituting k(x) with its explicit expression, we arrive at logf-DxS) = @log(^-) + logi-DiSi)

Exponentiating, we have : DiS D1S1

/ 5

N 0

,

186

Global Versus Local Dependence between Random

Variables

i.e.

DlS _ ~ se

D& se

>

and integrating with respect to a;, we obtain 5

i-e

s

i-e

for some j(y). Using the symmetry of 0 , one can also write ci-e

—

ci-©

= ^

+

m{x)

for some m(x). Thus : 5 1 - 0 = S J - 0 + Sl2'e + const. Evaluating the constant at the point (0,0), we have its value to be - 1, hence we thus obtained the required formula. - In the Clayton family, 6 is equal to a constant greater than one, but in many other families with positive dependence, Q(x,y) is a decreasing function of x or y. This situation seems to be more realistic. Indeed, for small value of x, the event {X = x} is more informative as far as the variable Y is concerning that the vague event {X > x}. In that case, /12 is almost equal to hy, the marginal hazard, but h2/i is probably very large, and therefore Q(x,y) is large. As x increases the information provided by the event (X > x) and (X = x) becomes equivalent and hence 0 decreases to one. For example, let X = A + B and Y = A + C, with A, B, C being independent, with the same exponential standard exponential distribution r ( l , 1). In that case : |y - a;I + 2 exp{-min{x,y)) \y — x\ + 1 — exp(—min(x,y)) As x approaches infinity, 0 decreases from ^v-p to 1. - If 0 decreases to one when x or y tend to +00, then the dependence of the pair (X,Y) is at least £>TP(0,1) or DTP(1,0), i.e. h2/i{x,y) a n d h>i/2{x,y) decrease in x (respectively in y). Indeed: since 0 is greater than 1, h,2(x,y) is decreasing with x, and so is

Local Indices of Dependence

187

li2/i(x,y), therefore the dependence is at least DTP(0,1). Since G is symmetric in x and y, the dependence is also DTP(1,0). -Oakes [163], utilizing Eq. (6.17) and censored data if available, proposes an estimator of 0 when the data are discretized. (4) A generalization : A multivariate definition of 0 . Starting from Equation (6.17), one can straightforwardly generalize the definition of 0 in case of more than two variables by defining :

U W

_ 5(x)Dt^5(x) DkS(t)DjS{ii)

where x = (xi,X2,-..,xm) and Dj = -^-. As in the case of two variables, it is the ratio of two hazards . If, for example X = (Xi,X-2, ...,Xm) is the random vector associated with the failures of members of a cluster or a family, then 0(x) is the ratio of — ^ L^', the hazard of the member j given that the member k failed at Xk = xk while the other members survived beyond Xi for i = 1,2,...,j - l,j + l,...,k -l,k + l,...,n, and - D ' s ^ , the hazard of member j given Xi > Xi for i = 1,2, ...,j — l,j + 1, ...,n. In the case of bivariate frailty models Oakes [163] has shown (see below Section 6.5.2 ) that 0(x) can be written as a function of the survival function S only : 0(x) = 0*(5(x)). In that case, Cho Paik et al. [45] have extended this definition to the case of multivariate frailty models allowing to take into account the failures of two (or more) members of the cluster in the hazard of a member. For example :

l

m

W

)

_

DtDmSMDjDnSix)

is the ratio of two hazards: the hazard of the member j given that the member k and the member m have failed while the other members have not, and the hazard of the member j given that the member m have failed while the other members have not. Cho Paik et al. [45] have also shown that in case of this model there are recursive relations between the functions 0*(5(x)), 0*(D m 5(x)), Q*{DmDkS(x)), etc. More exactly, if u =
188

Global Versus Local Dependence between Random

Variables

T,j
where ipW (u) denotes the j t h derivative of
(y(2)(u))2

•

There are recursive relationships among 0( 1 )(u),0( 2 )(u), ...,0( n )(u):

^>( u )0 ( n + 1 ) ( w ) = 2^ 1 ) ( U ) -

±g^.

That is a choice of the distribution of the frailty variable determines all the dependence functions between the members of the cluster. 6.3.8.4

Other measures derived from 0

To provide good estimators of dependence measures in case of censored data, Fan et al. [67] propose to use weighted dependence measures independent of the censoring distribution. For example, instead of Q(x,y) = h (x v)h(x v)' ^ n e y P r o P o s e to use the weighted dependence measure : D,

x=

IQ fp Hu,v)dudv So Jo hi(u,v)Ii2(u,v)dudv

H{x,y) hi(u,v)h2(u,v)dudv (6.22) where the instantaneous hazards are replaced by the cumulative hazards. More generally, the weighted dependence measures is of the form : =

J* /0y

, _ fi ft (c) = c) and w is a weight function equal to DiSDiS in Eq. (6.22). The weight function is chosen such that it does not depend on the censoring distribution. The 4> function is chosen to avoid no integrable

Local Indices of Dependence

189

differential function in the denominator which leads to estimators defined only on the grid formed by the uncensored failure times. Shih and Louis [207] have proposed other weighted measures based on the covariance process of the martingale residuals (see Section 6.3.8.1) for testing independence of bivariate survival data. 6.3.8.5

A local measure of LRD dependence : 7/

(1) Definition of 7/ : Holland and Wang [98] defined 7 / = $j$£ = Vlogf, where /(.,.) is a bivariate density. They show that this index is a continuous version of a(i, j), the log of a cross-ratio between four adjacent cells in an ordinal contingency table, given by :

a(i,j)

=log(-^

f-—).

Ji+l,jJiJ+l

Specifically : 7 JV /(*,v)= Mm - J - I n f ^ ^ + f^ + ^ V ' dx^o,dy^o dxdy \f(x + dx,y)f(x,y+ dy)J

(6.23)

(2) The joint distribution : - 7/ is symmetric. - 7/ is marginal free. Namely 7/ = j c , where c is the density of the associated copula. - If (X, Y) has a bivariate normal distribution with the correlation coefficient p , then 7/ = 3 ^ 5 - In this case this index is a constant. Conversely if 7/ is a constant, it is straightforward to show that the density f(x,y) is of the form a(x; 0)b{y\ 9)exp{9xy). See Jones [119]. (3) Dependence : Evidently, if the components of the pair (X, Y) are independent, 7 / = 0 . Equation (6.23) enables us to deduce that the condition 7/(x,y)>0,

Vx,Vy

190

Global Versus Local Dependence between Random

Variables

is equivalent to f(x, y) being TP2. Hence 7/(2;, y) is an appropriate index for measuring local LRD dependence. Jones [118] has shown, using a kernel method, that 7/(2:0,2/0) is a local version of the linear correlation coefficient. 6.4

Non-parametric Estimation of Local Indices

To estimate I or 0 , we ought to estimate the bivariate hazard h(x,y), the two conditional hazards hi(x,y) and h,2{x,y). 6.4.1

The univariate

case : estimation

and

of H(x)

We shall briefly survey estimation of H(x) and proceed to the bivariate case at hand. In the univariate case, with a random sample (JCi, X2,..., Xi,..., Xn) we associate the ordered sample (X^,X^)^ • • • i-^COi • • • >-^(n))- An empirical estimate of H(x) - the cumulative hazard - is the Nelson-Aalen [162] estimate :

Hn(x)= Y, X(i) *—'< X

nn — — ».i + 1

To obtain a smooth estimate of h(x), one considers h(x) as the derivative of H(x) and the smoothing is obtained by means of a convolution-product with a kernel K. Specifically :

K{x)

= JKbn(x =

- u)dHn(u)

l — Y K({X'X{i))). nbn f-f n-i + 1 \ bn

(6.24) (6.25) I

Here, bn is the window width, if is a bounded mapping from R to R with K(x)dx = 1, satisfying lim \x\K(x) = 0.

Non-parametric

Estimation

191

of Local Indices

To assure that the obtained estimate hn(x) is consistent, the window width has to satisfy, lim bn = 0 n—>oo

and lim nbn = oo . n—>oo

A measure of discrepancy between the estimator hn(x) from the actual hazard h(x) is the mean square error at x : MSEx{h~n) = E(hj})

- h{x)f .

The right-hand side term can be decomposed into two terms : bias and variance . MSEx{hn)

= [Eh^[x) - h{x)}2 +

varhn(x).

A kernel K which is asymptotically optimal is the Epanechnikov kernel

It assures that the mean integrated square error (MISE) MISE

= E{ f{h(x)

-

h(x))2dx)

will be minimum. In the univariate case the theoretical optimal choice for the window width, when the criterion is MISE, is bn oc n ~ s . 6.4.2

Bivariate

case and conditional

The bivariate hazard H(x,y) Aalen estimate [l], [162] :

hazards

is estimated by an analogue of the Nelson-

Hn(x,y)=

^2 {xi,yi)

n - i+1

:xi<x,yi
In this case hn(x,y) can also be obtained using a convolution-product by means of a bivariate kernel , KbXn,b2n{u,v) : hn{x,y)

=11

Kbln,b2„(x - u,y - v)Hn(du,dv)

.

192

Global Versus Local Dependence between Random

Variables

The two conditional hazards hi(x,y), i — 1,2, are obtained in the same manner from the partial cumulative hazards. For example :

(zt.j/i)

:yi
Hence hi(x,y)

=

Kbln(x-u)Hi(du,y). J[0,x]

6.4.2.1

Consistency and asymptotic normality of the estimate of h(x,y)

Dabrowska et al. [53] have established conditions to assure consistency and asymptotic normality of these estimators, in a general case where the data are censored. If the data are not censored, and if the following conditions are valid : (1) compact support : {x,y) e J — [si,s2] x [ti,t2] Q J e where e > 0 is small and J £ = [st - e, s2 + e] x [h - e, t2 + e] C [0, T] with S(T) > 0. (2) conditions on h(x,y) : (a) h(x, y) is bounded and strictly positive : 0 < inf{h(x,y)

: (x,y) € J J < sup{h(x, y) : (x,y) € Je} < oo.

(b) h(x,y) is twice continuously differentiate in

(x,y).

(3) on the kernel : - K(x,y) is a function of bounded variation on [—1,1]2, vanishing on the boundary of this set,

K(u)du = 0 ,

fuu'K(u)du

= d(K)I

where / is the identity matrix, and d(K) is a finite constant, and c(K) = f K*(u)du. (4) on the bandwidth : There exist sequences bin, i = 1,2, which satisfy

Non-parametric

Estimation

of Local Indices

193

(a) max(bin,b2n) ->• 0, (b) nb\nb2n -> oo (c) logn = o(nblnb2n) are valid. Then hn(x,y) is a consistent estimator of h(x,y), asymptotically normal, with an asymptotic mean square error (AMSE) at almost all points [x, y) given by :

The first term on the right-hand side is the variance, the second is the bias. One observes that at the upper tail of the distribution, where S(x,y) is small, the variance of the estimator is large. 6.4.2.2

Consistency and asymptotic normality of the two conditional hazards

Conditions for consistency and asymptotic normality of the conditional hazards are similar to those for the bivariate hazards. There are as follows: (1) compact support : Let (x,y) £ Ju = [s\ - c,s2 + e] x [0,t2], and J 2e = [0,*i] x [ti e, t2 + e] where Jie C [0, r] with S(T) > 0; (2) on hi(x,y) : (a) hi(x, y), i = 1, 2 is bounded and strictly positive ; (b) 0 < mi{hi(x,y) : (x,y) £ Jie} < sup{h(x,y) : (x,y) £ Jie) < oo ; (c) hi(x,y) i = 1,2 is twice continuously differentiable with respect to the ith coordinate of (x,y). (3) on the kernel : - K(x) is a function of a bounded variation on [—1,1], vanishing on the boundary of this set, - JuK(u)du = 0, f u2K(u)du = di(K), a finite constant, and 2 d(K) = / K (u)du. (4) on the bandwidth : There exist sequences <2jn, i = 1,2 which satisfy : - max(ain) —> 0, naj„ —> oo - logn = o(nain).

194

Global Versus Local Dependence between Random

Variables

If these conditions are fulfilled, then for almost all (x,y) in J^, hin(x,y) i = 1,2, is a consistent estimator of hi(x,y), asymptotically normal, with an asymptotic mean square error (AMSE) at the point (x, y) given by :

6.4.3

Estimation

of the indices I and 0

Having estimators for h(x,y), hi(x,y) and h,2(x,y), we can obtain estimators for l(x,y) and Q(x,y). The consistency and asymptotic normality of l(x,y) and Q(x,y) are obtained using the delta method . If we choose an optimal bandwith for hi(x,y) and h,2(x,y), then the rate of convergence for hi(x,y), and h2(x,y) will be n - 4 / 5 , which is faster than the rate for h(x,y), which is n - 2 / 3 . Namely only the variance of h(x,y) contributes to the variance of l(x,y) and ®(x,y). In other words varl(x,y) 6.5

«

varh(x,y).

A Search for the Localisation of the Maximal Association

We shall illustrate this search by means of three families of Archimedean copulas previously studied in the chapter on copulas. Here, we shall use the definition from the survival functions. (1) The Clayton family : This family verifies 0 = const. As we shall see that yields a "late" dependence : S(x,y)

= ((Sxix))1-"

+ ( 5 2 ( y ) ) 1 - a ) 1 / a - 1 , a > 1.

(2) The Gumbel-Hougaard family : S(x,y)

= exp [- ( ( - l n ( 5 i ( ; r ) ) ) 1 / a + ( - l n ( 5 2 ( y ) ) ) 1 / a ) a ] , 0 < a < 1.

In this case 0(s) = 1 + (1 - a ) / ( - a l n ( s ) ) and 0 decreases from +co to 1 when S decreases from 1 to 0. This property characterizes an "early" dependence. (3) The Frank family :

A Search for the Localisation of the Maximal

Association

195

Fig. 6.5 Densities with uniform marginals and T = 0.243, a) Clayton's model b) Gumbel-Hougaard model.

S{x,y) = logQ [l + (a

(a^1}

;

j ,0 < a < 1.

In this case the index 0 (s) = - s / 1 _gi) is decreasing from Zn (7) / (7 1) to 1 when S decreases from 1 to 0. Here the dependence is "median". We represent the three families for the same r, i.e. the same global dependence. Note the peak in the case of Clayton model near the point (1,1), while for the Gumbel-Hougaard model it is at (0,0). Note also a saddle in the vicinity of (1/2,1/2) for Frank's model.

196

Global Versus Local Dependence between Random

Fig. 6.6

6.5.1

Variables

Frank's density, r = 0.243, and uniform marginals.

Lower and upper tail dependence butions

for the three

distri-

These notions have been thoroughly investigated by Joe [108] and Nelsen [160]. Above we have defined these three copulas by means of their survival functions S, Si and 52- Therefore employing the notations of Section 4.2 of Chapter 4, we shall utilize copulas C'(u, v), recalling that the upper tail dependence of C'(u, v) is the lower tail dependence of C(u, v) and vice versa. For example, if we are searching for the upper tail dependence then: C'(u,u) C(u,u) o — lim„-n= lifflu-to • 1- u u However C(u, u) = 0

,. y" 1 (2^) = limt-+oo TTT— •

u

ip(t)

For the Clayton family
A Search for the Localisation of the Maximal Association

197

The Clayton family therefore possesses an upper tail dependence, which we shall call late dependence. For the Gumbel-Hougaard and Frank families , we find 5 — 0. In these cases the dependence is not late. Computing the lower tail dependence we obtain : ,. However C{u, u) = l-2u

C'(u,u) u

C(u,u) 1—u

+ C(u, u) = 1 - 2u + ip-1 (2
1- u

1 - if

1

(t)

For the Gumbel-Hougaard family
0 and the remaining

dependence

Oakes [163] has shown that for Archimedean copulas, 0 is a function of S, and that this property is characteristic. Specifically : B(x,y)=Q*(S(x,y)) where
ip'(s) and v?"(s) being the first and second derivative of (p. We can, therefore, easily represent the "evolution" of 0 for the three families discussed above.

As it was indicated above for the Gumbel-Hougaard family, 0(s) = 1 + zif^;, which decreases from infinity to 1 as S decreases from 1 to zero. The large values of 0 when 5 is in the vicinity of one, correspond to an "early" dependence for this family. We observe from Figure 6.7 that 0 is the largest for the Hougaard family when S > 0.75.

198

Global Versus Local Dependence between Random

Variables

Hougaard /

/

-Frank

Clayton

I ' ' ' ' I i ' i i I i i i i I i i i i I i ' i i [ i ' ' i I ' ' i i I ' ' i '

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

I

0.9

Fig. 6.7 Evolution of 0 as a function of S for the Clayton, Hougaard and Frank families (T = 0.243).

For the Frank family, 0 decreases from a finite value ^ f to 1 when S decreases from 1 to zero. For this family we notice that 0 is the largest for the intermediate values of S (S between 0.7 and 0.5). In this family the dependence is therefore "median".

A Search for the Localisation of the Maximal Association

199

Fig. 6.8 Evolution of the indices f(x, x) and l(x, x) as a function of x for the 3 families with r = 0.243.

For the Clayton family, 0 is constant, and hence 0 is the highest among the three distributions when S is small. From the figure 6.7 the corresponding values of S are from 0.5 to 0. The dependence is "late". Oakes [163] has shown that ^ ^ can be interpreted as a conditional r, and in particular for the Clayton family the ratio of the concordant and discordant pairs is constant regardless of the value of S. This, of course, applies also to "late" dependence. 6.5.3

Index 7 and the instantaneous

dependence

As it was shown in Section 6.3.8.5, the index 7^ measures local LRD dependence. We describe this property for the three distributions. For the Clayton family, jf(x,x) is maximal for x near to 1, conversely -ff(x,x) is maximal for x near to zero for the Hougaard family and the Frank family occupies intermediate position. Note that the conditional covariance rate does not indicate in a simple manner the localization of the dependence.

200

Fig. 6.9

6.5.4

Global Versus Local Dependence between Random

Variables

0 ( x , x ) according to x r = 0.243, a) Clayton's model b) Hougaard's model.

Simulation estimation

of the survival of 0

bivariate

distributions

and

We shall finally illustrate briefly the appropriateness of the local indices to retrieve local dependence in bivariate survival distribution. This is accomplished by simulating the three families of distributions Clayton's, Hougaard's and Frank's which served as models of different modes of dependence. The reader is referred to Section 4.10 in the chapter on Copulas for some details. We have used the method described by Genest [81] to simulate the distributions of Clayton's and Frank's families and the method described by Lee [141] to simulate the Hougaard family. We shall use the Dabrowska's method (Section 5) to estimate 0 . Figures 6.9 and 6.10 provide 0 and the 0.90-confidence limit obtained by the jacknife method on the diagonal y = x. Variations of 0 as a function of x allow us to retrieve the shape of the dependence.

A Search for the Localisation of the Maximal

Fig. 6.10

Association

@(x,x) as a function of x, T = 0.243, Frank's family.

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Index

conditionally increasing CI, 39, 40, 43, 61 in sequence CIS, 39, 40, 43, 61 cone order association, 54 convex function, 40, 60, 61, 84, 91, 95 copula, 65-112, 162, 184 Ali-Mikhail-Haq's, 72, 75, 81, 87 Archimax, 65, 94-96 Archimedean, 65, 84, 94, 95, 102, 110, 118, 160, 194 Clayton's, 60, 86, 94, 111, 186, 194 diagonal, 68, 70, 89-92 discontinuity, 65 dual of, 68, 70 Farlie-Gumbel-Morgenstern's FGM, 70, 75, 87, 88, 114, 118 iterated, 71 Frank's, 59, 78, 86, 102, 111, 195 Gumbel-Hougaard's, 86, 94, 96, 111, 177, 194 m-dimensional Archimedean, 102-104 normal copula, 155 piecewise additive, 97-98 piecewise quadratic, 98-101 Plackett's, 72, 75, 106 polynomial, 74, 75, 118 counting process, 178 covariance, 151-153

Abel criterion, 87 asymptotic dependence, 177 asymptotic independence, 177 asymptotic mean square error AMSE, 193, 194 Bayesian methodology, 63 beta function, 83, 141 binomial negative distribution, 81 bivariate exponential distribution, 178 bivariate extreme value distribution, 177 bivariate Gumbel exponential distribution, 122 bivariate inverted Dirichlet distribution, 166, 170 bivariate normal distribution BVN, 65, 71, 76, 153, 155, 189 bivariate Pareto distribution, 154, 166 Blomqvist's medial coefficient, 161 canonical correlation coefficient, 150, 165 Cauchy's distribution, 152 censoring, 179, 183, 187, 188, 192 Chebyshev's identity, 53 coherent system, 50 comonotone, 61 conditional covariance rate, 179-183 217

218 Q-covariance, 151 covariance function, 178-179 cumulative marginal hazard, 178 delta method, 174, 194 density weighting function, 77 Dirac's measure, 70 directionally convex function, 60 Dirichlet's distribution, 45, 46, 48 distortion function, 78 domain of attraction, 95, 96 entropy, 149, 150, 163-170 exchangeability, 56 exponential distribution, 42, 122, 140 extreme value distribution, 43, 83, 87, 94, 95, 144 Farlie-Gumbel-Morgenstern distribution FGM, 113, 147 iterated, 119 filtration, 178 Frechet's bound, 55, 68-70, 87 lower, 55, 77, 86, 87, 101, 102, 108, 158, 177 upper, 55, 69, 72, 77, 80, 82, 86, 87, 155, 158, 162, 181, 184 frailty model, 78-84, 102, 179, 187 correlated gamma, 82 frailty variables, 65 gamma distribution, 45, 80, 86 incomplete gamma function, 134 independence, 7-11, 16-19 information, see entropy, 149 information matrix, 168 interaction, 180, 182

Index Gasser-Miiller's, 174 Laplace's transform, 79-81, 86, 87, 95, 102, 112 latent variable, 63 left tail decreasing LTD, 38, 42 linear correlation coefficient, 11-13, 16, 22, 24-27, 29, 30, 116, 118, 124, 131-136, 149, 153-157, 162, 164, 173, 175, 189, 190 Ling's theorem, 88, 90 Lorentz inequality, 61 marginal distribution, 65 Marshall-Olkin's construction, 80, 112 martingale, 178 max-infinitely divisible distribution, 83 mean integrated square error MISE, 191 mixture distribution, 80, 83 multinomial distribution, 32, 45, 46, 48 multivariate hypergeometric distribution, 45, 46, 48 multivariate normal distribution, 40, 52 multivariate totally positive of order 2 MTP 2 , 40, 43 nearest-neighbor method, 174 negative dependence, 43-50, 86 association, 32, 46-48 lower orthant NLOD, 44, 45, 47, 48 quadrant, 46 superadditive, 48-50 upper orthant NUOD, 44, 45, 47, 48 Nelson-Aalen's estimator, 190, 191

jacknife method, 200 Kendall's r , 149, 159-162 kernel, 190, 191 Epanechnikov's, 174, 191

odds-ratio, 72, 180, 184 order statistic, 140-141, 144 ordering association, 56

Index

concordance, 61 directionally convex, 61 DTP(0,1), 58, 60, 86 integral stochastic, 55, 60-61 LRD, 58, 59 PDD, 57 positive dependence, 32 PRD, 56 stochastic, 32 supermodular, 60-61 T P 2 , 58, 61 overdispersion, 37 Polya function of order 2 PF2, 42, 45, 48 Poisson distribution, 45 positive dependence, 31, 33-186 association, 36-37, 39, 42, 43, 54 by mixture, 41-42 by total positivity DTP(m,n), 40-41, 54 function dependence, 37, 54, 57 likelihood ratio LRD, 39, 40, 42, 54, 86, 87, 119, 157, 159, 190, 199 lower orthant PLOD, 35-40 lower set PLSD, 36 quadrant PQD, 34-35, 39, 42, 54, 60, 157, 170 regression PRD, 38-54 through the stochastic ordering PDS, 39 upper orthant PUOD, 35, 36, 40 upper set PUSD, 36 prior distribution, 63 quantile function, 151 Riischendorf's method, 73, 113 regularly varying at infinity, 95 Renyi's correlation ratio, 175 reverse regular of order 2 RR2, 44, 119 in pairs, 44, 45, 48

219

right corner set increasing RCSI, 41 right tail increasing RTI, 38, 42 Schweizer-Wolff's index, 159 self-consistent estimators, 119 setwise dependence, 32, 50-54 association SA, 52 by mixture, 53 negatively, 53 positively lower orthant SPLOD, 50, 52 positively upper orthant SPUOD, 50, 52 positively upper set SPUSD, 50, 51 setwise independence, 51 Spearman's ps, 149, 157, 158, 161, 162 squeeze algorithm, 100, 101 stable distribution, 80, 87 stochastic dependence, 32 supermodular function, 35, 60 tail dependence, 83, 177-178 lower, 31, 43, 196-197 upper, 31, 43, 196-197 total positivity T P 2 , 39-41, 81, 119 in pairs, 40 TPoc, 81 truncation invariance, 66 Weibull distribution, 122-130

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meteorological phenomena in nature and interdependence

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'' 10 medical, social, and political aspects of our existence, not to mention the economic structures, are too numerous to be cited individually. Moreover, the

dependence is obviously not deterministic but of a stochastic nature. However, it seems that none of the departments of statistics, engineering, economics and mathematics in the academic institutions throughout the world offer courses dealing with dependence concepts and measures. This book can thus be viewed as an attempt to remedy the situation, and it has been written for a graduate course or a seminar on correlation and dependence concepts and measures. A modest background in mathematical statistics and probability and integral calculus is required. The book is not a full-scale expedition up another statistical Alp. Rather, it is a tour over a somewhat neglected but important terrain. The chapter on correlation is written for a layman.

"... it should be on the shelf of any statistician who must deal with

dependence." Biometrics

"The monograph should be useful to any person who does research in the area of dependence, and to any person who may wish to correctly apply positive dependence notions and ordering* in practice." Mathematical Reviews

Imperial College Press www.icpress.co.uk