A characterization of admissible linear estimators of fixed and random effects in linear models

Metrika (2008) 68:157–172 DOI 10.1007/s00184-007-0149-0

A characterization of admissible linear estimators of fixed and random effects in linear models Ewa Synówka-Bejenka · Stefan Zontek

Received: 10 January 2006 / Published online: 24 August 2007 © Springer-Verlag 2007

Abstract In the paper the problem of simultaneous linear estimation of fixed and random effects in the mixed linear model is considered. A necessary and sufficient conditions for a linear estimator of a linear function of fixed and random effects in balanced nested and crossed classification models to be admissible are given. Keywords Linear model · Linear estimation · Linear prediction · Admissibility · Admissibility among an affine set · Locally best estimator 1 Introduction The problem of characterization of linear admissible estimators in general linear model has been extensively examined in the literature. At first Cohen (1966) characterized admissible linear estimators of the mean vector within Gauss–Markov model with covariance matrix σ 2 I. Rao (1976) considered model for which the mean vector is varying through a linear subspace and covariance matrix has the form σ 2 V with V known. Further generalizations were given by Mathew et al. (1984), Klonecki and Zontek (1988) and Baksalary et al. (1995). The problem of unbiased linear estimation of fixed and random effects was considered by Henderson (1973), Harville (1976) and further generalizations were given by Rao (1987). Robinson (1991) extensively presents the history of BLUP and its relevance to the foundations of statistics (see references therein).

E. Synówka-Bejenka · S. Zontek (B) Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. prof. Z. Szafrana 4 a, 65-516 Zielona Góra, Poland e-mail: [email protected] E. Synówka-Bejenka e-mail: [email protected]

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We are interested in admissible linear estimation of fixed and random effects in linear models in which the set of covariance matrices is finitely generated. In considered models the class of admissible linear estimators contains unbiased estimators (the best linear predict does not exist). To give a characterization of linear admissible estimators (unbiased estimators) we reduce the problem to linear estimation of the fixed effects only in some linear model. Basic for us are LaMotte’s results LaMotte (1982, 1997) concerning admissibility in a general linear model without any restrictions on the parametric space. The first result gives necessary and sufficient conditions for a linear estimator to be admissible and the second one shows that each admissible linear estimator is the limit of unique locally best estimator at points in the set being a special extension of the original parameter set. Throughout this paper, Mm×t denotes the space of m × t real matrices. The symbols A
, R(A) and N (A) stand for the transpose, column space and null space of A ∈ Mm×t , respectively. For A ∈ Mm×m we define the linear operator T At mapping Mm×t into Mm×t for every m × t matrix B by T At (B) = AB. The image of T At is denoted by R(T At ). For A1 ∈ Mm 1 ×t1 , . . . , Ak ∈ Mm k ×tk the symbol diag(A1 , . . . , Ak ) denotes the matrix which the diagonal consists of A1 , . . . , Ak . 2 Preliminaries Let Y be a random n-vector having the following structure Y = Xβ + Z 1 u 1 + · · · + Z k u k + e, where β is a p-vector of unknown parameters (fixed effects), X ∈ Mn× p , Z 1 ∈ Mn×m 1 , . . . , Z k ∈ Mn×m k are known nonzero matrices, while u 1 ∈ Rm 1 , . . . , u k ∈ Rm k , e ∈ Rn are unobservable random vectors. It is assumed that u 1 , . . . , u k (random effects) and e (random errors) are uncorrelated random vectors with zero means and 2 I , respectively. It is convenient to rewrite the variances σ12 Im 1 , . . . , σk2 Im k and σk+1 n above model as follows Y = Xβ + Z u + e, where Z = (Z 1 , . . . , Z k ) and u = (u
1 , . . . , u
k )
. Clearly, E(Y ) = Xβ and cov(Y ) = 2 I , where D = cov(u) = diag(σ 2 I , . . . , σ 2 I ). This will be scheZ D Z
+ σk+1 n 1 m1 k mk matically written as 2 In ). (1) Y ∼ (Xβ, Z D Z
+ σk+1 We are interested in estimation of θ = [(K
Xβ)
, (Q
1 Z 1 u 1 )
, . . . , (Q
k Z k u k )
]


(2)

in the class of linear estimators L
Y = (L 0 , L 1 , . . . , L k )
Y, where K , L 0 ∈ Mn×t0 , . . . , Q k , L k ∈ Mn×tk . To compare estimators we use the ordinary quadratic risk function

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159

2 R(L
Y ; β, σ12 , . . . , σk+1 )



= E (L Y − θ ) (L Y − θ ) 

= tr (L 0 − K )
Xββ
X
(L 0 − K ) + L
0 cov(Y )L 0 +

k 

L i
Xββ
X
L i +

i=1

+

k 

k 

L i
cov(Y )L i

i=1

σi2 (L i

− Qi )




Z i Z i
(L i

− Qi ) −

i=1

k 

 σi2 L i
Z i Z i
L i

.

i=1

On the base of a partial ordering in the set of risk functions we define a partial ordering in the set of linear estimators. Namely, we say that L
Y is better than M
Y , if 2 2 R(L
Y ; β, σ12 , . . . , σk+1 ) ≤ R(M
Y ; β, σ12 , . . . , σk+1 ) 2 and strict inequality holds for at least one combination of these for all β, σ12 , . . . , σk+1 parameters. An estimator L
Y is said to be admissible for θ if there is no other estimator in E = {N
Y : N ∈ Mn×(to +···+tk ) } better than L
Y . Since in estimated function random effects are also considered, the risk function has the different structure than the risk function of linear estimator of fixed effects only. Therefore we cannot use directly results concerning a general linear model.

2.1 The dual problem We show that the problem of characterization of admissible linear estimators in model (1) is equivalent to a problem of characterization of admissible estimators in other linear model with some restrictions to a class of linear estimators considered in new model. Let Y be a random (k + 1)n-vector having the following structure 
 Y = Y
, (Z 1 u 1 )
, . . . , (Z k u k )
. From (1) we get that

 Y∼

Xβ,

k+1 

σi2 Vi

,

(3)

i=1

where  
X = X
, 0, . . . , 0 , Vi = (v1 + vi+1 )(v1 + vi+1 )
⊗ Z i Z i
, i = 1, . . . , k,

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and Vk+1 = v1 v1
⊗ In , while vi is the i-th versor in Rk+1 and the symbol ⊗ denotes the Kronecker product. As an equivalent of an estimator L
Y of θ in model (1), we take ⎡

L0 L1 ⎢ 0 −Q 1 ⎢ L
Y = ⎢ . .. ⎣ .. . 0 0

⎤
· · · Lk ··· 0 ⎥ ⎥ .. ⎥ Y .. . . ⎦ · · · −Q k ⎡

K ⎢0 ⎢ and we consider it as an estimator of K
Xβ, where K = ⎢ . ⎣ ..

0 0 .. .

··· ··· .. .

⎤ 0 0⎥ ⎥ .. ⎥ , in the model .⎦

0 0 ··· 0

(3). Since

L
Y − θ = L
Y −K
Xβ,

(4)

we have that
 E (L
Y − θ )
(L
Y − θ ) = E{[L
Y − K
Xβ]
[L
Y − K
Xβ]}, so the risk function of L
Y can be considered as the risk function of the linear estimator L
Y of K
Xβ in the above model. A class of considered linear estimators of K
Xβ in model (3) is restricted to the set E o = {N
Y : N ∈ Lo }, where t0 +···+tk Lo = L o + R(T
), o

while L o = diag(0, −Q 1 , . . . , −Q k ) and
o = v1 v1
⊗ In . Introducing on E o relation “better than” induced by the partial ordering on the set of risk functions of L
Y in E o (the same set, which induces relation “better than” in E), we define that L
Y ∈ E o is admissible for K
Xβ among Lo if L ∈ Lo and there is

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no other estimator in E o better than L
Y . Since the set of risk functions of estimators in E o coincides with the set of risk functions of estimators in E, we get the following lemma. Lemma 2.1 A linear estimator L
Y of θ is admissible in model (1) if and only if the corresponding estimator of K
Xβ is admissible among Lo in the model (3). In the dual problem a linear function of the expectation of Y is estimated, so we can apply results concerning admissible linear estimation in a general linear model. 2.2 Some known results Following LaMotte (1982) we consider (W1 , W2 ) = (cov(Y ) , EY EY
) as an argument of the risk function of L
Y . Namely, R(L
Y ; (W1 , W2 )) = E{[L
Y − K
Xβ]
[L
Y − K
Xβ]}
 = tr L
W1 L + (L − K )
W2 (L − K ) . In next sections we will assume that β is a real number, i.e., that p = 1, so the set   2 T = { cov(Y ), EY EY
: β ∈ R, σ12 ≥ 0, . . . , σk+1 ≥ 0} is finitely generated closed convex cone. To characterize admissible estimators of K
Xβ we use stepwise procedure elaborated by LaMotte (1982). Recall that an estimator L
Y is said to be locally best at a point W ∈ T among L, where L is an affine subset of Lo , if L ∈ L and R(L
Y ; W ) ≤ R(M
Y ; W ) for every M ∈ L. It is known (see LaMotte 1982) that an estimator L
Y of K
Xβ is locally best at a point W = (W1 , W2 ) ∈ T among L iff
(W1 + W2 )L =
W2 K , t0 +···+tk where
is a symmetric ((k +1)n ×(k +1)n)-matrix such that L = L +R(T
).
Denote by B(W |L) the set of these matrices L in L for which the estimator L Y is locally best at W among L. Note that B(W |L) is an affine set, which can be written in following way

B(W |L) = {L +
1 M : M ∈ M(k+1)n×(to +···+tk ) },
 where
1 =
I(k+1)n − (
(W1 + W2 )
)+ (
(W1 + W2 )
) . A point W ∈ T is called trivial if B(W |L) = L. Note also that for a point (W1 , W2 ) in T the set B((W1 , W2 )|L) contains only one point iff R(
(W1 + W2 )
) = R(
). For W ∈ T an estimator L
Y in E o for which B(W |Lo ) ={L} will be called unique locally best

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(ULBE for short) at W ∈ T among Lo . Of course ULBE is admissible among Lo (see LaMotte 1982, Theorem 3.6). When T is finitely generated closed convex cone, a theorem of LaMotte (1982), which is the basis of stepwise procedure, can be written in the following form (see also Azzam et al. 1988). Theorem 2.1 Under adopted assumptions, an estimator L
Y is admissible for K
Xβ among L if and only if there exist a nontrivial point W ∈ T such that L ∈ B(W |L) and L
Y is admissible for K
Xβ among B(W |L). To characterize admissible estimators for K
Xβ among Lo in model (3) we use also a connection between admissible estimators and limits of uniquely best linear estimators given by LaMotte (1997) (see also St¸epniak 1987; Zontek 1988). For model (3) a result of LaMotte (1997) is given below. Theorem 2.2 Each linear estimator of K
Xβ admissible among Lo is a limit of members of Lo that are uniquely best among Lo in T . To obtain a characterization of admissible linear estimators for a linear function of β and u, we use the following modification of Shinozaki’s lemma (Shinozaki, 1975) under restriction to E o (see Klonecki and Zontek 1988, Proposition 2.4). t0 +···+tk Lemma 2.2 If L
Y is admissible for K
Xβ among L o + R(T
), then for any o

matrix C ∈ M(t0 +···+tk )×s the estimator C
L
Y is admissible for C
K
Xβ among s ). L o C + R(T
o

3 Main results In this section we consider the k-way balanced nested classification random model and k-way balanced crossed classification random model, being the special cases of model (1). We obtain explicit formulae for all admissible linear estimators of K
Xβ among Lo in model (3). So by Lemma 2.1 we get a characterization of linear estimators admissible for (2) in models considering in this section. 3.1 The k-way nested classification Let Yi1 ...ik+1 , where i j = 1, 2, . . . , n j for j = 1, . . . , k + 1, be a random variable having the following structure Yi1 ...ik+1 = β + u 1i1 + u 2i1 i2 + · · · + u ki1 ...ik + ei1 ...ik+1 , where β is an unknown parameter, u 1i1 , . . . , u ki1 ...ik and ei1 ...ik+1 , are uncorrelated 2 , respectively. random variables with zero means and variances σ12 , . . . , σk2 and σk+1 Under these assumptions arraying the Yi1 ...ik+1 in lexicon order in n-vector Y , where n = n 1 n 2 · · · n k+1 , we get k-way nested classification model being a special case

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of model (1) for which X = 1n 1 ⊗ · · · ⊗ 1n k+1 = 1n and Z i = In 1 ⊗ · · · ⊗ In i ⊗ 1n i+1 ⊗ · · · ⊗ 1n k+1 , i.e.,  Y ∼ 1n β,

k 

σi2 (In 1

⊗ · · · ⊗ In i ⊗ Jn i+1

2 ⊗ · · · ⊗ Jn k+1 ) + σk+1 In

,

(5)

i=1

where 1a denotes the a-vector of ones while Ja = 1a 1a
. To simplify notation let Z 0 = 1n and Z k+1 = In . Putting pi = n i+1 · · · n k+1 for i = 0, . . . , k and pk+1 = 1 define E0 =

1 Z 0 Z 0
p0

and Ei =

1 1
Z i Z i
− Z i−1 Z i−1 for i = 1, . . . , k + 1. pi pi−1

Note that E 0 , . . . , E k+1 are idempotent and orthogonal matrices such that Z i Z i
= pi

i 

E j for i = 0, . . . , k + 1.

(6)

j=0

To characterize admissible estimators L
Y for K
Xβ among Lo in model (3) corresponding to model (5) we prove the following lemma, which gives necessary and sufficient conditions for an estimator L
Y to be ULBE at the point W = (W1 , W2 ) ∈ T .   k+1
Lemma 3.1 An estimator L
Y is ULBE at a point i=1 si Vi , s0 X X in T among Lo in model (3) corresponding to model (5) if and only if s0 ≥ 0, . . . , sk ≥ 0, sk+1 > 0 and ⎡ ⎤ L0 L1 · · · Lk ⎢ 0 −Q 1 · · · 0 ⎥ ⎢ ⎥ L=⎢ . .. . . .. ⎥ , . ⎣ . . . ⎦ . 0 0 · · · −Q k where s0 p 0 L 0 = k+1 E0 K , j=0 s j p j ⎛ ⎞ i  s p i i Li = ⎝ E j ⎠ Q i for i = 1, . . . , k. k+1 s p l l l= j j=0

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Proof An estimator L
Y is locally best at s j ≥ 0 for j = 0, . . . , k + 1 and
o

k+1 

 k+1

i=1 si Vi , s0 X X






in T among Lo iff

si Vi + s0 X X




L = s0
o X X
K .

i=1

In more details this equation can be written as M L 0 = s0 X X
K , M L i − si Z i Z i
Q i = 0 for i = 1, . . . , k, k+1 si Z i Z i
. Of course, the above equations have only one solution, where M = i=0 with respect to L 0 , . . . , L k iff the matrix M is nonsingular, that is iff sk+1 > 0. The assertion follows from (6) by noting that E 0 , . . . , E k+1 are idempotent and orthogonal matrices.   Theorem 3.1 For an estimator L
Y of K
Xβ to be admissible among Lo in model (3) corresponding to model (5) it is necessary and sufficient that L belongs to the set ⎧⎡ L 0 (a0 ) L 1 (a0 , a1 ) ⎪ ⎪ ⎪ ⎨⎢ 0 −Q 1 ⎢ ⎢ .. .. ⎪ ⎣ . . ⎪ ⎪ ⎩ 0 0

⎫ ⎤ · · · L k (a0 , . . . , ak ) ⎪ ⎪ ⎪ ⎬ ⎥ ··· 0 ⎥ ⎥: ai ∈ [0, 1], i = 0, . . . , k , .. .. ⎪ ⎦ . . ⎪ ⎪ ⎭ ··· −Q k

(7)

where L 0 (a0 ) = a0 E 0 K and ⎧ ⎨

⎤ ⎫ ⎡ i−1  i−1 ⎬  ⎣ (1 − al )⎦ E j Q i for i = 1, . . . , k. L i (a0 , . . . , ai ) = ai E i + ⎩ ⎭ j=0

l= j

Proof The necessary condition. Let L belong to the set (7) with ai ∈ [0, 1) for i = 0, . . . , k. Define si =

pi

k

ai

j=i (1 − a j )

for i = 0, . . . , k and sk+1 =

1 pk+1

.


Using  k+1 Lemma 3.1
it can be verify that L Y is ULBE at the point W = i=1 si V i , s0 X X in T . Since for any fixed values ai+1 ∈ [0, 1), . . . , ak ∈ [0, 1)

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we have that si runs over [0, +∞) when ai ∈ [0, 1) for i = 0, . . . , k, the set given by (7) is the closure of {M : M
Y is ULBE at a point in T among Lo }. The first part of the proof is completed by Theorem 2.2. Now assume that L belongs to the set describe by (7). By Theorem 2.1 it is sufficient to construct a sequence of points W 1 , . . . , W l in T such that L1 = B(W 1 |Lo ), . . . , Ll−1 = B(W l |Ll−1 ) ={L}. Number l depends on number of these coefficients a0 , a1 , . . . , ak which are equal 1. Let I = {i : ai = 1; i = 0, . . . , k}, let m1 =

k + 1, min I,

when I = ∅, otherwise

t0 +···+tk ). This means that L = L0 . Let W = (W1 , W2 ) be a and let L = L + R(T
0 point in T defined by

W =

⎧ ⎪ ⎪ 0, ⎪ ⎪ ⎪  ⎪ ⎨ 1

1 p0

 X X
,

a0
p1 V1 , p0 (1−a0 ) X X

 ⎪ ⎪ ⎪ m 1 −1 ⎪ ⎪ ⎪ ⎩

i=1 pi

ai m 1 −1 j=i



(1−a j )

when m 1 = 0, ,



Vi +

1 pm 1

Vm 1 ,

p0

a0 m 1 −1 j=0

(1−a j )

X X
,

when m 1 = 1, otherwise.

Further part of the proof, we describe in the the form of an algorithm. Label1: Since
(W1 + W2 )L =
W2 K , hence L
Y is locally best at W among L. if m 1 = k + 1; In this case R(
(W1 + W2 )
) = R(
), that is B(W |L) ={L}. So L
Y is admissible among Lo . else; Let B(W |L) be a new L. The new L can be represented as t0 +···+tk L = L + R(T
),

where 
=

v1 v1


⊗ In −

m1 

Ei .

i=0

Let I \ {m 1 } be a new I, let m2 =

k + 1, min I,

when I = ∅, otherwise

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E. Synówka-Bejenka, S. Zontek

and let W = (W1 , W2 ) be a point in T defined by  ⎧ 1 ⎪ V , 0 , ⎪ m ⎨  pm 2 2 m W = 2 −1 ai ⎪ ⎪ Vi + m 2 −1 ⎩ p (1−a ) i=m 1

i

j=i

j

1 pm 2

Vm 2 , 0 ,

when m 2 = m 1 + 1, otherwise.

Finally let m 2 be a new m 1 . goto label 1; endif;   Note that such procedure must stop at finite number of steps, since in each step the dimension of the corresponding affine set decrease. In extreme case, when all a0 , . . . , ak are equal 1, we must do (k + 1) + 1 steps. Remark 3.1 An admissible estimator L
Y is unbiased for K
Xβ in model (3) corresponding to model (5) if and only if L belongs to the set (7) with a0 = 1. 3.2 The k-way crossed classification Let Yi1 ...ik+1 , where i j = 1, 2, . . . , n j for j = 1, . . . , k + 1, be a random variable having the following structure Yi1 ...ik+1 = β + u 1i1 + · · · + u kik + ei1 ...ik+1 , where β is an unknown parameter, u 1i1 , . . . , u kik and ei1 ...ik+1 , are uncorrelated random 2 , respectively. Using variables with zero means and variances σ12 , . . . , σk2 and σk+1 these assumptions and arraying the Yi1 ...ik+1 in lexicon order in n-vector Y , where n = n 1 n 2 · · · n k+1 , we obtain so-called the k-way crossed classification random model, which is given by  Y ∼ 1n β,

k 

2 σi2 (Jn 1 ⊗ · · · ⊗Jn i−1 ⊗ In i ⊗Jn i+1 ⊗ · · · ⊗Jn k+1 )+σk+1 In

.

(8)

i=1

Let p0 = n, Z 0 = 1n , pk+1 = 1 and Z k+1 = In . Putting pi = i = 1, . . . , k and defining 1 Z 0 Z 0
, p0 1 Ei = Z i Z i
− E 0 for i = 1, . . . , k, pi
− (E 0 + · · · + E k ) E k+1 = Z k+1 Z k+1 E0 =

123

 j=i

n j for

A characterization of admissible linear estimators of fixed and random effects in linear models

167

we have that E 0 , . . . , E k+1 are idempotent and orthogonal matrices and that Z 0 Z 0
, . . . ,
are linear combinations of them. Z k+1 Z k+1 To give the necessary and sufficient conditions for a linear estimators L
Y to be admissible estimators for K
Xβ among Lo in model (3) corresponding to model (8) we prove the following lemma.   k+1
in T among s V , s X X Lemma 3.2 An estimator L
Y is ULBE at a point i i 0 i=1 Lo in model (3) corresponding to model (8) if and only if s0 ≥ 0, . . . , sk ≥ 0, sk+1 > 0 and ⎡ ⎤ L0 L1 · · · Lk ⎢ 0 −Q 1 · · · 0 ⎥ ⎢ ⎥ (9) L=⎢ . .. .. ⎥ , .. ⎣ .. . . . ⎦ 0

0

· · · −Q k

where s0 p 0 L 0 = k+1 E0 K , j=0 s j p j  si pi si pi + sk+1 Li = E 0 Q i for i = 1, . . . , k. E i + k+1 si pi + sk+1 j=0 s j p j Proof Let



k+1
i=1 si V i , s0 X X



belong to T that is s0 ≥ 0, . . . , sk+1 ≥ 0. As in the

proof of Lemma 3.1 an estimator L
Y with L given by (9) is ULBE iff sk+1 > 0 and L 0 = s0 M −1 X X
K , L i = si M −1 Z i Z i
Q i for i = 1, . . . , k, where M=

k+1  i=0

si Z i Z i


=

k+1  i=0

si pi

E0 +

k 

(si pi + sk+1 ) E i + sk+1 E k+1 .

i=1

  Theorem 3.2 For an estimator L
Y of K
Xβ to be admissible among Lo in model (3) corresponding to model (8) it is necessary and sufficient that L belongs to the set ⎧⎡ L 0 (a0 ) L 1 (a1 , A1 ) ⎪ ⎪ ⎪ ⎨⎢ 0 −Q 1 ⎢ L∈ ⎢ . .. ⎪ ⎣ .. . ⎪ ⎪ ⎩ 0 0

⎫ ⎤ · · · L k (ak , Ak ) ⎪ ⎪ ⎪ ⎬ ⎥ ··· 0 ⎥ ⎥ : ai ∈ [0, 1], i = 0, . . . , k , .. .. ⎪ ⎦ . . ⎪ ⎪ ⎭ ··· −Q k

(10)

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E. Synówka-Bejenka, S. Zontek

where L 0 (a0 ) = a0 E 0 K , L i (ai , Ai ) = ai [E i + (1 − a0 )Ai E 0 ]Q i for i = 1, . . . , k and

1 1−ai 1 j=1 1−a j − (k

A i = k

− 1)

for a1 ∈ [0, 1), . . . , ak ∈ [0, 1),

otherwise A1 , . . . , Ak are nonnegative numbers such that where I = {i ∈ {1, . . . , k} : ai = 1} .

k i=1

Ai =



i∈I

(11) Ai = 1,

Proof The necessary condition. Let L belong to the set (10) with ai ∈ [0, 1) for i = 0, . . . , k. Define a0 s0 = (1 − a0 ) p0

 k  i=1

ai ai + 1 , si = (1 − ai ) (1 − ai ) pi

for i = 1, . . . , k, sk+1 = 1.


 it can be check that L Y is ULBE at the point W = Using Lemma 3.2 k+1
i=1 si Vi , s0 X X in T . Now we shown that the set given (10) is the closure of

{L : L
Y is ULBE at a point in T among Lo }. At first note that for any fixed values a1 ∈ [0, 1), . . . , ak ∈ [0, 1) we have that s0 runs over [0, +∞) when a0 ∈ [0, 1) and that every si ∈ [0, +∞) when kai ∈ [0, 1) Ai ≤ k. for i = 0, . . . , k. Of course for Ai defined by (11) we have that 1 < i=1  1 Since kj=1 1−a → +∞ when a → 1 for at least one j, i.e., I  = ∅, we have that j j k / I. So by Theorem 2.2, the first part of the i=1 Ai → 1 and that A j → 0 for j ∈ proof is completed. The sufficient condition can be established as follows. (1) If every ai ∈ [0, 1) for i = 0, 1, . . . , k, then one can easily check that L
Y is ULBE at  k  i=1

ai a0 Vi + Vk+1 , (1 − ai ) pi (1 − a0 ) p0

 k  i=1

ai
+ 1 XX ∈ T (1 − ai )

among Lo and of course is admissible for K
Xβ. (2) If a0 = 1, then we have that X X
L = X X
K , so L
Y is locally best at (0, X X
) ∈ T among Lo , but it is not unique. The set B((0, X X
)|Lo ) can be represented in the following way t0 +···+tk L1 = L + R(T
), 1

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169

where
1 = v1 v1
⊗ (In − E 0 ). Next, if I = ∅, then it is easy to verify that
1 W1 L = 0, where W1 =  k ai
i=1 (1−ai ) pi Vi + Vk+1 . Hence L Y is locally best at (W1 , 0) ∈ T and since R(
1 W1
1 ) = R(
1), therefore L
Y is admissible among Lo .    1 1 If I = ∅, then taking i∈I pi Vi , 0 ∈ T we have that
1 i∈I pi Vi L = 0, which means that L
Y is locally best at this point among L1 , but it is not unique.  The affine set B ( i∈I p1i Vi , 0)|L can be expressed as t0 +···+tk ), L2 = L + R(T
2

where !
2 =

v1 v1




⊗ In − E 0 +



" .

Ei

i∈I

 Finally, if we take the point (W1 , 0) ∈ T , where W1 = i∈I (1−aaii ) pi Vi + Vk+1 , we can easy to check that
2 W1 L = 0. Since R(
2 W1
2 ) = R(
2 ), we get that L
Y is admissible among Lo . (3) Assume that a0 ∈ [0, 1) and I = ∅. Let I1 = {i ∈ I : Ai > 0} and let I2 = I \I1 .  Of course Ai = 1. Taking i∈I1

⎛

⎞  Ai a 0 W =⎝ Vi , X X
⎠ , pi (1 − a0 ) p0 i∈I1

we have that ⎛

⎞  Ai a0 a0
o ⎝ Vi + X X
⎠ L =
o X X
K . pi (1 − a0 ) p0 (1 − a0 ) p0 i∈I1

So L
Y is locally best at W among Lo and B(W |Lo ) can be represented as t0 +···+tk L1 = L + R(T
), 1

where ⎡
1 = v1 v1
⊗ ⎣ In − (E 0 +



⎤ E i )⎦ .

i∈I1

123

170

E. Synówka-Bejenka, S. Zontek

    1
Vi , 0 ∈ T we get that
1 pi Vi L = 0, so L Y is locally i∈I2 i∈  I2 1 best at this point among L. The affine set B ( pi Vi , 0)|L1  = {L} can be

Taking

 

1 pi

i∈I2

expressed as t0 +···+tk L2 = L + R(T
), 2

where ! v1 v1



2 =

⊗ In − (E 0 +



" Ei ) .

i∈I

It can be checked that
2

 i∈I

ai (1−ai ) pi

 Vi + Vk+1 L = 0 and that L is unique

solution of this equation. This implies that L
Y is admissible among Lo , which ends the proof.   Note that, distinct from model (5) in model (8) it is enough to do at most three steps, because for this model not only a number of the coefficients ai , which are equal 1, but mainly the form of coefficients Ai determine a number of steps. Remark 3.2 An admissible estimator L
Y is unbiased for K
Xβ in model (3) corresponding to model (8) if and only if L belongs to the set (10) with a0 = 1. 3.3 The one-way classification For k = 1 models (5) and (8) reduce to so called one-way balanced random model with n 1 cells and with n 2 observations in each cells, for which Yi1 i2 = β + u 1i1 + ei1 i2 , where i 1 = 1, 2, . . . , n 1 , i 2 = 1, 2, . . . , n 2 . In matrix notation it can be written as   (12) Y ∼ 1n β, σ12 (In 1 ⊗ Jn 2 ) + σ22 In . From Lemma 2.1 and Theorem 3.1 (or Theorem 3.2) we get the following corollary. Corollary 3.1 An linear estimator (L 1 , L 2 )
Y is admissible for [(K
X β)
, (Q
1 Z 1 u 1 )
]
in model (12) if and only if (L 1 , L 2 ) ∈ {[L 0 (a0 ), L 1 (a0 , a1 )] : a0 ∈ [0, 1], a1 ∈ [0, 1]},

(13)

L 0 (a0 ) = a0 E 0 K

(14)

where

123

A characterization of admissible linear estimators of fixed and random effects in linear models

171

and L 1 (a0 , a1 ) = a1 [E 1 + (1 − a0 )E 0 ]Q 1 .

(15)

We are interested in admissibility of linear estimators for β + u 1i1 in model (12). At first we present a characterization of admissible linear estimators of [β, u 1i1 ]
. Lemma 3.3 An estimator 
Y is admissible for [β, u 1i1 ]
in model (12) if and only if  ∈ {(l0 (a0 ), l1 (a0 , a1 )) : a0 ∈ [0, 1], a1 ∈ [0, 1]}, where l0 (a0 ) =

(16)

a0 1n n

(17)

# $ a1 a0
(In 1 ⊗ 1n 2 ) − 1n 1n 1 vi1 , l1 (a0 , a1 ) = n2 n1

and

(18)

while vi1 is the i 1 -th versor in Rn 1 . Proof It is enough to put K = n1 1n and Q =

1 n 2 (In 1

⊗ 1n 2 )vi1 in Corollary 3.1.

 

Using Lemma 3.3 and Lemma 2.2 with C = (1, 1)
we get the following corollary. Corollary 3.2 An estimator 
Y is admissible for β + u 1i1 in model (12) if and only if 
Y = a0 Y + a1 (Y i1 . − a0 Y ) with a0 ∈ [0, 1] and a1 ∈ [0, 1], where Y i1 . =

1 n2

n 2

i 2 =1 Yi 1 i 2

and Y =

1 n

n 1

i 1 =1

n 2

i 2 =1 Yi 1 i 2 .

When we take C = (1, 0)
we get the well-know characterization of admissible linear estimators for β. Corollary 3.3 An estimator 
Y is admissible for β in model (12) if and only if 
Y = a0 Y with a0 ∈ [0, 1]. Acknowledgments The authors are grateful to the referees for their useful comments and remarks, which substantially improved the paper.

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