Admissibility of Estimated Regression Coefficients Under Generalized Balanced Loss
There are some discussions concerning the admissibility of estimated regression coefficients under the balanced loss function in the general linear model. We study this issue for the generalized linear regression model. First, we propose a generalized weighted balance loss function for the generaliz...
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Hong-Bing Qiu Ji Luo Jiajia Zhang 2020-02-13T12:51:28Z 2020-02-13T12:51:28Z 2015 Admissibility of Estimated Regression Coefficients Under Generalized Balanced Loss / Hong-Bing Qiu, Ji Luo, Jiajia Zhang // Український математичний журнал. — 2015. — Т. 67, № 1. — С. 128–134. — Бібліогр.: 16 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165434 519.21 There are some discussions concerning the admissibility of estimated regression coefficients under the balanced loss function in the general linear model. We study this issue for the generalized linear regression model. First, we propose a generalized weighted balance loss function for the generalized linear model. For the proposed loss function, we study sufficient and necessary conditions for the admissibility of the estimated regression coefficients in two interesting linear estimation classes. Ведуться дєякі дискусії щодо допустимості оцінєних коєфіцієнтів регресії для збалансованої функції втрат у загальній лінійній моделі. В роботі вивчається ця проблема для узагальненої лінійної моделі регресії. Так, запропоновано узагальнену зважену функцію втрати балансу для узагальненої лінійної моделі. Для вказаної функції втрат ми вивчаємо необхідні та достатні умови допустимості оцінених коефіцієнтів регресії для двох цікавих лінійних випадків оцінювання. Supported in part by the National Natural Science Foundation of China (11171058), the National Social Science Foundation of China (13CTJ012), Guangdong Provincial Natural Science Foundation of China (S2012040007622), Zhejiang Provincial Natural Science Foundation of China (LQ13A010002), and the National Statistical Science Research Project (2012LY129) en Інститут математики НАН України Український математичний журнал Статті Admissibility of Estimated Regression Coefficients Under Generalized Balanced Loss Допустимість оцінених коефіцієтів регресії при узагальненій збалансованій втраті Article published earlier |
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Admissibility of Estimated Regression Coefficients Under Generalized Balanced Loss |
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Admissibility of Estimated Regression Coefficients Under Generalized Balanced Loss Hong-Bing Qiu Ji Luo Jiajia Zhang Статті |
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Admissibility of Estimated Regression Coefficients Under Generalized Balanced Loss |
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Admissibility of Estimated Regression Coefficients Under Generalized Balanced Loss |
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Admissibility of Estimated Regression Coefficients Under Generalized Balanced Loss |
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Admissibility of Estimated Regression Coefficients Under Generalized Balanced Loss |
| title_sort |
admissibility of estimated regression coefficients under generalized balanced loss |
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Hong-Bing Qiu Ji Luo Jiajia Zhang |
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Hong-Bing Qiu Ji Luo Jiajia Zhang |
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2015 |
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Допустимість оцінених коефіцієтів регресії при узагальненій збалансованій втраті |
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There are some discussions concerning the admissibility of estimated regression coefficients under the balanced loss function in the general linear model. We study this issue for the generalized linear regression model. First, we propose a generalized weighted balance loss function for the generalized linear model. For the proposed loss function, we study sufficient and necessary conditions for the admissibility of the estimated regression coefficients in two interesting linear estimation classes.
Ведуться дєякі дискусії щодо допустимості оцінєних коєфіцієнтів регресії для збалансованої функції втрат у загальній лінійній моделі. В роботі вивчається ця проблема для узагальненої лінійної моделі регресії. Так, запропоновано узагальнену зважену функцію втрати балансу для узагальненої лінійної моделі. Для вказаної функції втрат ми вивчаємо необхідні та достатні умови допустимості оцінених коефіцієнтів регресії для двох цікавих лінійних випадків оцінювання.
|
| issn |
1027-3190 |
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https://nasplib.isofts.kiev.ua/handle/123456789/165434 |
| citation_txt |
Admissibility of Estimated Regression Coefficients Under Generalized Balanced Loss / Hong-Bing Qiu, Ji Luo, Jiajia Zhang // Український математичний журнал. — 2015. — Т. 67, № 1. — С. 128–134. — Бібліогр.: 16 назв. — англ. |
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2025-11-25T07:26:38Z |
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| fulltext |
UDC 519.21
Hong-Bing Qiu (Guangdong Univ. Technology, Guangzhou, China),
Ji Luo (Zhejiang Univ. Finance and Economics, Hangzhou, China),
Jiajia Zhang (Univ. South Carolina, Columbia, USA)
ADMISSIBILITY OF ESTIMATED REGRESSION COEFFICIENTS
UNDER GENERALIZED BALANCED LOSS*
ДОПУСТИМIСТЬ ОЦIНЕНИХ КОЕФIЦIЄНТIВ РЕГРЕСIЇ
ПРИ УЗАГАЛЬНЕНIЙ ЗБАЛАНСОВАНIЙ ВТРАТI
There are some discussions about the admissibility of estimated regression coefficients under the balanced loss function
in the general linear model. This paper studies this issue for the generalized linear regression model. First, we propose
the generalized weighted balance loss function for the generalized linear model. For the proposed loss function, the
paper investigates sufficient and necessary conditions for the admissibility of the estimated regression coefficients in two
interesting linear estimation classes.
Ведуться деякi дискусiї щодо допустимостi оцiнених коефiцiєнтiв регресiї для збалансованої функцiї втрат у загаль-
нiй лiнiйнiй моделi. В роботi вивчається ця проблема для узагальненої лiнiйної моделi регресiї. Так, запропоновано
узагальнену зважену функцiю втрати балансу для узагальненої лiнiйної моделi. Для вказаної функцiї втрат ми ви-
вчаємо необхiднi та достатнi умови допустимостi оцiнених коефiцiєнтiв регресiї для двох цiкавих лiнiйних випадкiв
оцiнювання.
1. Introduction. The general linear regression model is
y = Xβ + e, E(e) = 0, Cov(e) = σ2In, (1.1)
where y ∈ Rn is an observable random vector, X ∈ Rn×p is a known design matrix with full-column
rank, β ∈ Rp and σ2 > 0 are unknown parameters, and e is a random error vector. Let Rn and
Rn×m stand for the sets of n-dimensional column vectors and real (n×m)-matrices, respectively.
Considering goodness of fit, the estimating equation is
(y −Xβ̂)′(y −Xβ̂) = min
d
(y −Xd)′(y −Xd), (1.2)
where d is an any estimator of the regression coefficients β in the linear model (1.1). The solution of
(1.2), β̂ = (X ′X)−1X ′y, is called the Least Squares (LS) estimator of regression coefficients β. The
expression of (y−Xd)′(y−Xd) measures a kind of fitting goodness of estimation model. However,
statisticians also measure superiorities of parameter estimation with respect to its precision. One of
the popular loss functions is the quadratic loss
(d− β)′(d− β), (1.3)
which directly depicts the precision of parameter estimation d.
Considering both of the above criteria, Zellner [16] proposed the concept of balanced loss function
* Supported in part by the National Natural Science Foundation of China (11171058), the National Social Science
Foundation of China (13CTJ012), Guangdong Provincial Natural Science Foundation of China (S2012040007622), Zhejiang
Provincial Natural Science Foundation of China (LQ13A010002), and the National Statistical Science Research Project
(2012LY129).
c© HONG-BING QIU, JI LUO, JIAJIA ZHANG, 2015
128 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ADMISSIBILITY OF ESTIMATED REGRESSION COEFFICIENTS UNDER GENERALIZED BALANCED LOSS 129
LB(d;β, σ2|S) = w(y −Xd)′(y −Xd) + (1− w)(d− β)′S(d− β), (1.4)
where w ∈ [0, 1] is a real number and S ∈ Rp×p is some known positive definite matrix. It’s obvious
that the balanced loss function (1.4) can reflect not only statistical properties of estimator itself,
but also goodness of fit, which is more comprehensive when evaluating superiority of estimation,
compared with loss function (1.2) or (1.3). Under the balanced loss function (1.4), there has been
a number of studies about estimators for regression coefficients β. In [3], Dey et al. considered
estimators δ(y) = θ̂ + g(y), where θ̂ is the LS estimator, and y is in the action space, under the
quadratic loss and the balanced loss. This kind of studies are also can be seen in [10, 11]. In order
to estimate the exponential mean time to failure, the authors [13] introduced the weighted balanced
loss function, defined as
LWB(d;β, σ2|S) = wq(β)(y −Xd)′(y −Xd) + (1− w)q(β)(d− β)′S(d− β), (1.5)
where q(β) is any positive function of β, called the weight function. The related developments of
the weighted loss function were given by [1, 4, 14].
Meanwhile, there are many discussions about the admissibility of estimated regression coefficients
β in the linear model. First, we introduce the definition of admissible estimator. Let L(d;β, σ2) be a
given loss function, where d is an any estimator of β. R(d;β, σ2) = E[L(d;β, σ2)], is called the risk
function with respect to the loss function L(d;β, σ2). An estimator δ1 is said to dominate an estimator
δ2, if R(δ1;β, σ
2) ≤ R(δ2;β, σ
2) for all β, σ2, and the above inequality holds strictly for some β0,
σ20. An estimator δ is admissible with respect to the risk function R(d;β, σ2) if no other estimator
dominates it. Xu and Wu [15] studied the admissibility of estimated regression coefficients for β
in model (1.1) under the balanced loss function (1.4) and investigated its necessary and sufficient
conditions. In [8], Luo and Bai investigated the necessary and sufficient conditions for regression
coefficients’ unbiasedness, efficiency and admissibility in a general linear model. However, the classic
linear regression model (1.1) assumes that the random error is independent, while it is very common
to see correlated structure in practice, which can be described by the Gauss – Markov model. The
Gauss – Markov model can be expressed as
y = Xβ + e, E(e) = 0, Cov(e) = σ2V, (1.6)
where y ∈ Rn is an observable random vector, X ∈ Rn×p is a known design matrix with full-column
rank, β ∈ Rp and σ2 > 0 are unknown parameters, V ∈ Rn
≥ is a nonnegative definite matrix, e is a
random error vector. The admissibility of estimated regression coefficient in (1.6) was investigated
under quadratic loss function and matrix loss in [2]. From then on, there has been considerable
interest in this issue. In [12], Rao characterized admissible estimators of a linear parametric function
Sβ in the class of all linear estimators under the assumption that V is a positive definite matrix. In
[9], the authors extended the results of [12] to the situation where the covariance matrix is singular.
Later the problem was developed both in unconstrained case and constrained case for the unknown
parameters by [5 – 7] and so on.
In this paper we investigate the admissibility of estimated regression coefficients β in (1.6), under
the generalized weighted balanced loss function. First, the generalized weighted balance loss function
for generalized linear model is proposed in Section 2. In Section 3, we investigate the admissibility
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
130 HONG-BING QIU, JI LUO, JIAJIA ZHANG
of estimated linear coefficients in two linear estimation classes. Finally, concluding remarks are given
in Section 4.
2. Generalized weighted balanced loss function. Let Rn
> and Rn
≥ stand for the sets of n
positive definite matrices and n nonnegative definite matrices, respectively. For a matrix A, A′,
tr(A), rk(A), A+ and R(A) denote transpose, trace, rank, Moore – Penrose inverse, the range space
of column vectors of A, respectively. R(A
...B) represents the range space of column vectors of the
merged matrix of A and B. A⊥ denotes a matrix of maximum rank that satisfies A′A⊥ = 0. The
dimension and orthogonal complement subspace of linear space R(A) are denoted by dimR(A) and
N(A). For nonnegative definite matrices A and B, A < B stands for the positivity of matrix B−A.
A ≤ B stands for the nonnegativity of matrix B −A. That is to say, A ≤ B denotes there exists the
Löwner partial order for matrices A and B.
Considering both of goodness of fit and precision of estimation, we propose the generalized
weighted balance loss function for generalized linear model (1.6), which is
LGWB(d;β, σ2|U, S) = wq(β)(y −Xd)′T+(y −Xd) + (1− w)q(β)(d− β)′S(d− β), (2.1)
where w ∈ [0, 1], is a real number, q(β) is a positive weight function, T = V +XUX ′ ∈ Rn
≥, U is a
symmetric (p× p)-matrix such that R(T ) = R(V
...X), S ∈ Rp
>. It is worthwhile pointing out that the
estimator, minimizing the expression (y −Xd)′T+(y −Xd), is the best linear unbiased estimation
of β, also called the Gauss – Markov estimator. Then, we investigate the admissibility of estimated
regression coefficients β in generalized linear model (1.6) under the generalized weighted balanced
loss function (2.1).
Because β is the location parameter of the generalized linear model (1.6), we are interested in the
two linear estimation classes of β,
K0 = {Hy : H ∈ Rp×n}, the class of all homogeneous linear estimators,
K1 = {Hy + a : H ∈ Rp×n, a ∈ Rp}, the class of all linear estimators.
Specific condition of H and a will be discussed in order to guarantee the admissibility of the
estimators.
3. Main results. Let R(d;β, σ2) denote the risk function of estimator d under the generalized
weighted balanced loss function (2.1), that is to say, R(d;β, σ2) = ELGWB(d;β, σ2).
Theorem 3.1. Under model (1.6) and loss function (2.1), the necessary and sufficient condition of
Hy is a linear admissible estimator of β in estimation class K0 is that under the quadratic loss function
(1.3),Gy is an admissible estimator ofCβ in estimation class K0, whereG = B1/2(H−wB−1X ′T+),
C = (1− w)B−1/2S and B = wX ′T+X + (1− w)S.
Proof. Suppose Hy ∈ K0. Under loss function (2.1), the risk function is
R(Hy;β, σ2) = q(β)E[w(y −XHy)′T+(y −XHy) + (1− w)(Hy − β)′S(Hy − β)] =
= q(β)[β′(I −X ′H ′)[wX ′T+X + (1− w)S](I −XH)β+
+wσ2tr((I −XH)′T+(I −XH)V ) + (1− w)σ2tr(SHVH ′)] =
= q(β){β′(I −X ′H ′)[wX ′T+X + (1− w)S](I −XH)β−
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ADMISSIBILITY OF ESTIMATED REGRESSION COEFFICIENTS UNDER GENERALIZED BALANCED LOSS 131
−σ2w2tr{T+X[wX ′T+X + (1− w)S]−1X ′T+V }+ σ2wtr(T+V )+
+σ2tr{H ′ − wT+X[wX ′T+X + (1− w)S]−1}[wX ′T+X + (1− w)S]×
×{H − w[wX ′T+X + (1− w)S]−1X ′T+}V }.
Let B = kX ′T+X + (1− k)S > 0, L = H − kB−1X ′T+, then H = L+ kB−1X ′T+. Substituting
the expressions of B and H into R(Hy;β, σ2), we have
R(Hy;β, σ2) =
= q(β){β′[X ′L′ − (1− w)SB−1]B[LX − (1− w)B−1S]β − σ2w2tr(T+XB−1X ′T+V )+
+σ2wtr(T+V ) + σ2tr{(H ′ − wT+XB−1)B(H − wB−1X ′T+)V }} =
= q(β){β′[X ′L′ − (1− w)SB−1]B[LX − (1− w)B−1S]β − σ2k2tr(T+XB−1X ′T+V )+
+σ2wtr(T+V ) + σ2tr(L′BLV )}.
If there exists My ∈ K0, superior to Hy, then
R(β, σ2, Hy)−R(β, σ2,My) ≥ 0
holds for all β ∈ Rp and σ2 > 0, and the equality does not always hold.
Let L̄ = M − wB−1X ′T+, equivalently M = L̄+ wB−1X ′T+. Then we obtain
R(Hy;β, σ2)−R(My;β, σ2) =
= q(β){β′[X ′L′ − (1− w)SB−1]B[LX − (1− w)B−1S]β − σ2w2tr(T+XB−1X ′T+V )+
+σ2wtr(T+V ) + σ2tr(L′BLV )− {β′[X ′L̄′ − (1− w)SB−1]B[L̄X − (1− w)B−1S]β−
−σ2w2tr(T+XB−1X ′T+V ) + σ2wtr(T+V ) + σ2tr(L̄′BL̄V )}} =
= q(β){β′[X ′L′ − (1− w)SB−1]B[LX − (1− w)B−1S]β + σ2tr(L′BLV )−
−β′[X ′L̄′ − (1− w)SB−1]B[L̄X − (1− w)B−1S]β − σ2tr(L̄′BL̄V )}.
For simplicity, denote Q = B1/2, then
R(Hy;β, σ2)−R(My;β, σ2) =
= q(β){β′[X ′L′Q− (1− w)SQ−1][QLX − (1− w)Q−1S]β + σ2tr[(QL)′(QL)V ]−
−β′[X ′L̄′Q− (1− w)SQ−1][QL̄X − (1− w)Q−1S]β − σ2tr[(QL̄)′(QL̄)V ]} =
= q(β){E(Gy − Cβ)′(Gy − Cβ)− E(Ḡy − Cβ)′(Ḡy − Cβ)},
where G = QL, Ḡ = QL̄, C = (1− w)Q−1S. Thus
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
132 HONG-BING QIU, JI LUO, JIAJIA ZHANG
R(Hy;β, σ2)−R(My;β, σ2) ≥ 0
holds for all β ∈ Rp and σ2 > 0, and the equality does not always hold, if and only if
E(Gy − Cβ)′(Gy − Cβ)− E(Ḡy − Cβ)′(Ḡy − Cβ) ≥ 0
holds for all β ∈ Rp and σ2 > 0, and the equality does not always hold. That is to say, as
an estimator of Cβ, Ḡy is superior to Gy under the quadratic loss function (1.3). Due to the
reversibility of Q = B1/2, Hy → Gy = QLy = Q(H − wB−1X ′T+)y is a one-to-one mapping
from K0 to itself. Thus under the generalized weighted balanced loss function (2.1), Hy is an
inadmissible estimation of β is equivalent to Gy is an inadmissible estimation of Cβ under quadratic
loss function (1.3). Equivalently, under the generalized weighted balanced loss function (2.1), Hy is
an admissible estimation of β if and only if Gy is an admissible estimation of Cβ under quadratic
loss function (1.3).
Theorem 3.1 is proved.
For the admissibility of Hy + a in estimation class K1, we can derive the following results.
Lemma 3.1. Under model (1.6) and loss function (2.1), R(Hy + a;β, σ2) and R(Hy;β, σ2)
satisfy the relation expression R(Hy + a;β, σ2) = R(Hy;β, σ2) + q(β)[a′Ba− 2a′B(I −HX)β],
where B is defined as Theorem 3.1.
Proof. We have
R(Hy + a;β, σ2) =
= q(β)E[w(y −X(Hy + a))′T+(y −X(Hy + a)) + (1− w)(Hy + a− β)′S(Hy + a− β)] =
= q(β){E[w(y −XHy)′T+(y −XHy) + (1− w)(Hy − β)′S(Hy − β)]+
+a′wX ′T+Xa+ a′(1− w)Sa− 2a′wX ′T+X(I −HX)β − 2a′(1− w)S(I −HX)β} =
= R(Hy;β, σ2) + q(β)[a′Ba− 2a′B(I −HX)β].
Now comprehensively considering Theorem 3.1, Lemma 3.1 and the above analysis, we can
obtain the result of Theorem 3.2.
Theorem 3.2. Under model (1.6) and loss function (2.1), the necessary and sufficient condition
of Hy+ a is an admissible estimator of β in estimation class K1 is that Hy is admissible in class K0
and a ∈ R(I −HX), where R(A) stands for the linear subspace generated by column vectors of A.
Proof. Firstly, we investigate the expression of a. According to Lemma 3.1, if the estimator with
the form Hy+ a is admissible, a must belong to R(I −HX). Otherwise, Hy+ (I −HX)γ will be
superior to Hy + a, where γ ∈ Rp. On the other side, if a ∈ R(I −HX), because β is unknown,
there is no another vector b, such that b′Bb− 2b′B(I −HX)β ≤ a′Ba− 2a′B(I −HX)β for all β,
and the inequality doesn’t hold for some β.
Remark 3.1. Theorem 2 of [15] investigated the admissibility of Hy+a in general linear model,
which is similar to Theorem 3.2. However, our proof simplifies the proving process greatly.
Theorems 3.1 and 3.2 clearly explain the admissibility under the weighted loss function and its
relationship with the quadratic loss. In the remaining of this paper, we will discuss the specific
situations when the estimated regression coefficient is admissible with respect to H and a. Firstly,
we rewrite Gauss – Markov model as
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 1
ADMISSIBILITY OF ESTIMATED REGRESSION COEFFICIENTS UNDER GENERALIZED BALANCED LOSS 133
y = X̃β̃ + e, E(e) = 0, Cov(e) = σ2V, (3.1)
where X̃ = XC−1, β̃ = Cβ,C is defined as Theorem 3.1.
Then we apply Theorem 3.4, Corollary 3.3 in [9] to model (1.6). For the purpose of clarity, we
restate the theorem and corollary here, which is referred as Lemmas 3.2 and 3.3 in this paper.
Lemma 3.2 [9]. Under model (1.6) and quadratic loss function (1.3), LY is admissible for β if
and only if L = M(X ′TX)−1X ′T, where M = LX satisfies
(a) M((X ′TX)−1 − I) ≥ 0;
(b) M((X ′TX)−1 − I) ≥M((X ′TX)−1 − I)M ′;
(c) R[(I −M)(X ′TX)(X ′TV )⊥] ⊂ R[(I −M)(X ′TX)(X ′TV )].
Lemma 3.3 [9]. Consider the setup described in Lemma 3.2, where X satisfies R(X) ⊂ R(V ).
Then LY is admissible for β if and only if L = M(X ′TX)−1X ′T, where M = LX satisfies
conditions (a) and (b) in Lemma 3.2.
Theorem 3.3. Under model (1.6) and loss function (2.1), Hy is a linear admissible estimator of
β in estimation class K0 if and only if G = MC(X ′T̃X)−1X ′T̃ , where M = GXC−1 satisfies
(a) M(C(X ′T̃X)−1C ′ − I) ≥ 0;
(b) M(C(X ′T̃X)−1C ′ − I) ≥M(C(X ′T̃X)−1C ′ − I)M ′;
(c) R[(I−M)(C ′−1X ′T̃XC−1)(C ′−1X ′T̃ V )⊥] ⊂ R[(I−M)(C ′−1X ′T̃XC−1)(C ′−1X ′T̃ V )],
where T̃ = V + X̃UX̃ ′, G = B1/2(H − wB−1C ′−1X ′T̃+), C = (1 − w)B−1/2S and B =
= wC ′−1X ′T̃+XC−1 + (1− w)S.
Theorem 3.4. Under model (1.6) and loss function (2.1), suppose R(X) ⊂ R(V ). Hy is a
linear admissible estimator of β in estimation class K0 if and only if G = MC(X ′T̃X)−1X ′T̃ ,
where M = GXC−1 satisfies conditions (a) and (b) in Theorem 3.3.
4. Concluding remarks. We extend the weighted balanced loss function to generalized weighted
balanced loss function and apply it to the Gauss – Markov model. It is worthwhile pointing out that
the weighted balanced loss function can not be applied to the Gauss – Markov model. Then we
considered the admissible estimator of regression coefficients under the generalized linear regression
model. Finally, we give the sufficient and necessary conditions for the admissibility among two linear
estimation classes. Theorems 3.1 and 3.2 described the relationship between two admissibility under
LGWB and the quadratic loss function (1.3) and Theorems 3.3 and 3.4 enabled us to verify if an
estimator is admissible.
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