114 A^VÇÚO 1n ò where y is an n 1 random vector of observations, X is a known n p matrix of full column rank, ε is an n 1 unobservable random vector,

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1 A^VÇÚO 1n ò 1Ï 2015c4 Chinese Journal of Applied Probability and Statistics Vol.31 No.2 Apr Optimal Estimator of Regression Coefficient in a General Gauss-Markov Model under a Balanced Loss Function Hu Guikai Peng Ping (School of Science, East China Institute of Technology, Nanchang, ) Abstract In this paper, we investigate optimal estimator of regression coefficient in a general Gauss- Markov model under balanced loss function. Firstly, necessary and sufficient conditions for linear estimators to be best linear unbiased estimator (BLUE) are provided. Secondly, we prove the best linear unbiased estimator is unique in the sense of almost everywhere, and also a balance between least squares estimator and optimal estimator under quadratic loss. Thirdly, loss robustness of the optimal estimator is discussed in terms of relative losses and relative saving losses. Finally, we give some conditions about the robust BLUE on the mis-specification of covariance matrix. Keywords: Optimal estimator, robustness, balanced loss function, general Gauss-Markov model. AMS Subject Classification: 62J12, 62H Introduction We open this section with some notations: Given a matrix A, the symbols M (A), A, A +, tr(a) will stand for the range space, the transpose, the Moore-Penrose inverse, the trace, respectively, of matrix A. The n n identity matrix is denoted by I n. For an n n matrix A, A > 0 means that A is a symmetric positive definite matrix. A 0 means that A is a symmetric nonnegative definite matrix, A B (A B) means that A B 0 (B A 0). R m n stands for the set composed of all m n real matrices. Consider the following linear model y = Xβ + ε, (1.1) E(ε) = 0, Cov (ε) = σ 2 V, The project was supported by the Natural Science Foundation of Jiangxi Province (20144BAB ), Humanities and Social Science Planning Foundation in College of Jiangxi Province (TJ1401) and the National Social Science Foundation of China (12BTJ014). Received August 14, doi: /j.issn

2 114 A^VÇÚO 1n ò where y is an n 1 random vector of observations, X is a known n p matrix of full column rank, ε is an n 1 unobservable random vector, V R n n is a known nonnegative definite matrix. Whereas, β R p and σ 2 > 0 are unknown parameters. This model is usually called a general Gauss-Markov model. For a comprehensive overview of this model, see Rao (1973). Considerable attention has been given in the past several decades to the problem of the best linear unbiased estimator of regression coefficient. For example, Rao (1972) proposed the unified theories of least squares. Albert (1973) obtained two kinds of expressions of the BLUE by using the general inverse of matrices. In their methods, they considered the goodness of fit of model and the precision of estimation, respectively. However, they did not take account of them together. Generally, either of the criterion of goodness of fit of model or precision of estimation is used to judge the performance of any estimator. Consequently, Considering both goodness of fit of model and precision of estimation together, Zellner (1994) has proposed the following balanced loss function(blf) L(d; β, σ 2 ) = θ(y Xd) (y Xd) + (1 θ)(d β) X X(d β), (1.2) where θ is a scalar lying between 0 and 1 which provides the weight assigned to the goodness of fit of the model, d is any estimator of β. The balanced loss function has received considerable attention in the literature under different setups. For example, Rodrigues and Zellner (1994) have used the balanced loss function in the estimation of mean time to failure. The aspects of preliminary test estimation and Stein-rule estimation under BLF were discussed by Giles et al. (1996), Ohtani et al. (1997), Ohtani (1998, 1999). Gruber (2004) obtained the empirical Bayes and approximate minimum mean square error estimators under a general balanced loss function and r uncorrelated linear models with random regression coefficients, and evaluated the efficiency of these estimators averaging over Zellner s balanced loss function. Jozani et al. (2006) motivated weighted balancedtype loss function and considered the issues of the admissibility, dominance, Bayesianity and minimaxity. Bansal and Aggarwal (2007, 2009, 2010) employed BLF to examine loss robustness of Bayes predictors of some finite population quantities. Since the best linear estimator does not exist in the class of the homogeneous linear estimators under the balanced loss risk, Hu and Peng (2010, 2011) studies the linear admissible estimators of the regression coefficient. In our brains, it is easy to think the problem whether the best linear unbiased estimator does not exist, and what properties has it when it exists? We now define the risk function as R(d; β, σ 2 ) = E[L(d; β, σ 2 )].

3 1Ï?m $±: ²ï e Gauss-Markov. 8Xê `O 115 Consider the following linear estimator classes: = {Ly : L R p n }. Definition 1.1 Ly is called an unbiased estimator of β, if E(Ly)=β for all β R p. Obviously, if Ly is an unbiased estimator of β, then LX = I p. If there exists a linear estimator Ly such that Ly is an unbiased estimator of β, then β is called a linearly estimable variable. We concern ourselves with the minimal risk properties of linear unbiased estimator of β. Therefore, we assume that β in the model (1.1) is a linearly estimable variable in the following paper. Definition 1.2 A linear unbiased estimator Ly is called a best linear unbiased estimator of β if for any linear unbiased estimator My in, R(Ly; β, σ 2 ) R(My; β, σ 2 ) holds for any β R p and σ 2 > 0. In this paper, we mainly discuss the best linear unbiased estimator of regression coefficient under the model (1.1) and the loss function (1.2), and obtain necessary and sufficient conditions for linear estimators to be BLUE. Furthermore, Loss robustness of BLUE in terms of relative losses and relative saving losses, and robust BLUE on the mis-specification of the covariance matrix are studied, respectively. The rest of this paper is organized as follows. In Section 2, we give some necessary and sufficient conditions for homogeneous linear estimators to be BLUE and obtain the unique BLUE of β in the sense of almost everywhere. Loss robustness of BLUE is placed in Section 3. The robust BLUE on the mis-specification of the covariance matrix is given in Section 4. Concluding remarks are placed in Section Best Linear Unbiased Estimator In this section, we provide some necessary and sufficient conditions for linear estimators to be a BLUE of β, and give the explicit expression of the BLUE. Theorem 2.1 In the model (1.1), Ly is a best linear unbiased estimator of β under the balanced loss (1.2) if and only if L satisfies the following conditions: where N X = I n X(X X) 1 X. LX = I p, (2.1) LV N X = θ(x X) 1 X V N X, (2.2)

4 116 A^VÇÚO 1n ò Proof Sufficiency: Assume that M y is a linear unbiased estimator of β, Since L satisfies condition (2.1), we have MX = LX, i.e., X (M L) = 0, then there exists a matrix Z R p n such that M = L + ZN X. By direct computation, we have R(My; β, σ 2 ) = E[θ(y XMy) (y XMy) + (1 θ)(my β) X X(My β)] = R(Ly; β, σ 2 ) + σ 2 tr[n X Z X XZN X V 2θN X Z X (I XL)V + 2(1 θ)n X Z X XLV ] = R(Ly; β, σ 2 ) + σ 2 tr(n X Z X XZN X V ) + 2σ 2 tr(x XLV N X Z θx V N X Z ), which combining with condition (2.2) will derive that R(My; β, σ 2 ) = R(Ly; β, σ 2 ) + σ 2 tr(n X Z X XZN X V ) R(Ly; β, σ 2 ), for all β R p and σ 2 > 0. Thus, Ly is the best linear unbiased estimator of β under the balanced loss function. Necessity: We suppose that Ly is the BLUE of β, then L satisfies the condition (2.1). Assume by contradiction that LV N X θ(x X) 1 X V N X. Take Z 0 = X XLV N X θx V N X, then we get Z 0 Z 0 = (X XLV θx V )N X Z 0 0. Thus we have tr[(x XLV θx V )N X Z 0 ] > 0. Hence, there exists a real number t < 0 such that σ2 tr[(x XLV θx V )N X tz 0 + tz 0N X (X XLV θx V ) + t 2 N X Z 0 X XZ 0 N X V ] < 0. Denote M = L + tz 0 N X, then My is a linear unbiased estimator of β, and R(My; β, σ 2 ) = E[θ(y XLy XtZ 0 N X y) (y XLy XtZ 0 N X y) + (1 θ)(ly + tz 0 N X y β) X X(Ly + tz 0 N X y β)] = R(Ly; β, σ 2 ) + σ 2 tr[(x XLV θx V )N X tz 0 + tz 0 N X (X XLV θx V ) + t 2 N X Z 0X XZ 0 N X V ] < R(Ly; β, σ 2 ), which is a contradiction to that Ly is the BLUE of β. Thus L satisfies the condition (2.2). The proof is completed. Theorem 2.2 In the model (1.1), denote L 0 = (X T + X) 1 X T + + θ(x X) 1 X V N X (N X V N X ) + N X, where T = V + XX. Then the following statements hold:

5 1Ï?m $±: ²ï e Gauss-Markov. 8Xê `O 117 (1) L 0 y is a BLUE of β under the balanced loss function (1.2). (2) L = {Ly : L = L 0 +Z[N X N X V N X (N X V N X ) + N X ], Z R p n } is an estimator class that contains all the best linear unbiased estimators. (3) If Ly L, then Ly = L 0 y holds almost everywhere, that is, L 0 y is the unique BLUE of β under the balanced loss function (1.2). Proof (1) It is easy to verify that L 0 satisfies conditions (2.1) and (2.2). Therefore (1) holds. (2) If L satisfies conditions (2.1) and (2.2). Then we have (L L 0 )X = 0, (L L 0 )V N X = 0, (2.3) and hence L L 0 = (L L 0 ) (L L 0 )X(X X) 1 X = (L L 0 )N X. (2.4) It follows from Equations (2.3) and (2.4) that 0 = (L L 0 )V N X = (L L 0 )N X V N X = (L L 0 )N X V N X (N X V N X ) + N X, which together with Equation (2.4) derives that L = L 0 + (L L 0 ) = L 0 + (L L 0 )N X = L 0 + (L L 0 )[N X N X V N X (N X V N X ) + N X ]. Suppose that Z = L L 0, then we have that L = L 0 + Z[N X N X V N X (N X V N X ) + N X ], that is, Ly = [L 0 + Z(N X N X V N X (N X V N X ) + N X )]y L. On the other hand, we suppose that Ly L, i.e., there exists a matrix Z such that L = L 0 + Z[N X N X V N X (N X V N X ) + N X ]. It is easy to verify that L satisfies conditions (2.1) and (2.2), i.e., Ly is the BLUE of β. Therefore L contains all the best linear unbiased estimators. (3) Suppose that Ly L, i.e., there exists a matrix Z R p n such that L = L 0 + Z[N X N X V N X (N X V N X ) + N X ]. Then we have E[(L L 0 )y] = E[Z(N X N X V N X (N X V N X ) + N X )y] = 0, D[(L L 0 )y] = D[Z(N X N X V N X (N X V N X ) + N X )y] = 0. Thus P{(L L 0 )y = 0} = 1, i.e., Ly = L 0 y holds almost everywhere, and hence L 0 y is the unique best linear unbiased estimator of β. The proof is completed. This theorem has given an expression of the BLUE, but it is not easy to analyze its properties on the basis of this. Therefore we must give another form in order to analyze it better. In the following, we first give two lemmas.

6 118 A^VÇÚO 1n ò Lemma 2.1 (Wang, 1987) In the model (1.1), y M (T ) holds almost everywhere. Lemma 2.2 (Yu and He, 1997) Let C and D be two real matrices, if D 0 and M (C) M (D), then (C D + C) + = C + DC + C + DN C (N C DN C ) + N C DC +. Theorem 2.3 In the model (1.1), denote L 1 = (X T + X) 1 X T + + θ(x X) 1 X V (T + T + X(X T + X) 1 X T + ), then L 1 y is the BLUE of β. Proof According to Lemma 2.2, we have (X T + X) 1 = X + T X + X + T N X (N X T N X ) + N X T X +, which together with Lemma 2.1 will yield that L 1 y = (X T + X) 1 X T + y + θ(x X) 1 X T T + y θ(x X) 1 X T (I N X )T + y + θ(x X) 1 X T N X (N X T N X ) + N X T (I N X )T + y = (X T + X) 1 X T + y + θ(x X) 1 X T N X (N X T N X ) + N X T T + y = (X T + X) 1 X T + y + θ(x X) 1 X V N X (N X V N X ) + N X y = L 0 y. Theorem 2.4 In the model (1.1), the best linear unbiased estimator L 1 y can be represented as θ β + (1 θ) β T, where β = (X X) 1 X y, βt = (X T + X) 1 X T + y. Furthermore, the minimal risk is R(L 1 y; β, σ 2 ) = σ 2 tr[(1 θ) 2 (X T + X) 1 X T + V X θ 2 (X X) 1 X V X + θv ]. Proof According to Lemma 2.1, by direct computation, we have L 1 y = (X T + X) 1 X T + y + θ(x X) 1 X V (T + T + X(X T + X) 1 X T + )y = (X T + X) 1 X T + y + θ(x X) 1 X T T + y θ(x X) 1 X X(X T + X) 1 X T + y = θ(x X) 1 X y + (1 θ)(x T + X) 1 X T + y. By the definition of R(d; β, σ 2 ), we have R(L 1 y; β, σ 2 ) = σ 2 tr[l 1X XL 1 V 2θXL 1 V + θv ] = σ 2 tr{[θ(x X) 1 X + (1 θ)(x T + X) 1 X T + ] X X[θ(X X) 1 X + (1 θ)(x T + X) 1 X T + ]V 2θX[θ(X X) 1 X +(1 θ)(x T + X) 1 X T + ]V + θv } = σ 2 tr[(1 θ) 2 (X T + X) 1 X T + V X θ 2 (X X) 1 X V X + θv ].

7 1Ï?m $±: ²ï e Gauss-Markov. 8Xê `O 119 This theorem shows that the best linear unbiased estimator of β under the balanced loss function (1.2) is a balance between the least-squares estimator and the best linear unbiased estimator under quadratic loss. Moreover, the weights assigned to the goodness of fit of model and the precision of estimation are consistent to the weights assigned to their corresponding optimal estimators. It is clear to illustrate the use of the balanced loss function (1.2). Corollary 2.1 Under the model (1.1), if V > 0, then the BLUE of β can be represented as θ β + (1 θ) β V, where β = (X X) 1 X y, β V = (X V 1 X) 1 X V 1 y. Since (X T + X) 1 X T + = (X V 1 X) 1 X V 1, if V > 0, it is easy to verify by Theorem 2.4. Therefore its proof is omitted here. 3. Loss Robustness In this section, we will examine the loss robustness of these estimators. It may be of interest to compare the risk of the BLUE of β with that of least squares estimator β and also with that of β T. By the definition of R(d; β, σ 2 ), we have 1 = R( β; β, σ 2 ) R(L 1 y; β, σ 2 ) = (1 θ) 2 σ 2 tr[(x X) 1 X V X (X T + X) 1 X T + V X], (3.1) 2 = R( β T ; β, σ 2 ) R(L 1 y; β, σ 2 ) = θ 2 σ 2 tr[(x X) 1 X V X (X T + X) 1 X T + V X]. (3.2) It is observed that 1 and 2 are non-negative. Furthermore, 1 is a decreasing function, whereas 2 is an increasing function of θ. In particular, For θ = 1/2, 1 = 2. The relative loss RL 1 in using β relative to L 1 y is RL 1 = and that in using β T relative to L 1 y is RL 2 = 1 R(L 1 y; β, σ 2 ) = 2 R(L 1 y; β, σ 2 ) = (1 θ)2 u 1 + θ(1 θ)u, (3.3) θ 2 u 1 + θ(1 θ)u, (3.4) where u = tr([(x X) 1 X V X (X T + X) 1 X T + V X]) tr[θv + (1 θ)(x T + X) 1 X T + V X θ(x X) 1 X V X]. (3.5) It is observed that RL 1 is maximized at θ = 0, whereas RL 2 is maximized at θ = 1. However, when θ = 1/2, RL 1 = RL 2 for all values of u. As u increases, RL 1 will approach

8 120 A^VÇÚO 1n ò (1 θ)/θ and RL 2 will approach θ/(1 θ). Zellner (1994) and Bansal and Aggarwal (2007, 2009, 2010) also observed similar behavior of relatives losses. In order to measure the relative increase in balanced loss risk by using the least squares estimator β instead of the BLUE with respect to the same by using β T, let us follow Bansal and Aggarwal (2009, 2010) to define relative savings loss (RSL) as RSL = 1 2 = ( 1 θ ) 2, which is also RL 1 /RL 2. Relative savings loss measures the balanced loss risk improvement over β that is sacrificed by the use of β T rather than the best linear unbiased estimator. It is interesting to observe that the RSL depends only on the weight parameter θ of the balanced loss function. Note that θ 1/2 implies RSL 1, which shows that β T is better than the least squares estimator β in the smaller balanced loss risk sense. 4. The Misspecification of the Covariance Matrix In this section, we consider the robustness of the best linear unbiased estimator against possible misspecification of the covariance matrix V. Assume that the adopted model is such that while the true model specifies that M 1 : y = Xβ + ε 1, E(ε 1 ) = 0, Cov (ε 1 ) = σ 2 V 1, M 2 : y = Xβ + ε 2, E(ε 2 ) = 0, Cov (ε 2 ) = σ 2 V 2. Assume that V 1 and V 2 are known nonnegative definite matrices. Let β BLUE i be the BLUE of β under model M i, respectively, i = 1, 2. Then β BLUE i = (X T i + X) 1 X T i + y + θ(x X) 1 X V i (T i + T i + X(X T i + X) 1 X T i + )y, (4.1) where T i = V i + XX, i = 1, 2. Lemma 4.1 estimators of zero vector: θ For linear model M 1 (or M 2 ), let U be the set of all linear unbiased U = {Dy : D R p n and E(Dy) = 0, for all β R p }. Then a linear unbiased estimator Ly is the BLUE of β under balanced loss (1.2) if and only if E(y L X XDy θy XDy) = 0, for all Dy U.

9 1Ï?m $±: ²ï e Gauss-Markov. 8Xê `O 121 Proof Sufficiency: Since Ly is a linear unbiased estimator of β, then F = L + MN X, M R p n, for arbitrary linear unbiased estimators F y. Denote D = MN X, then F = L + D. It is obvious that Dy is a linear unbiased estimator of zero vector. On the other hand, if Dy is arbitrary linear unbiased estimator of zero vector, then X D = 0, that is, D = MN X, M R p n. By E(y L X XDy θy XDy) = 0, we have R(F y; β, σ 2 ) = E[θ(y XF y) (y XF y) + (1 θ)(f y β) X X(F y β)] = R(Ly; β, σ 2 ) + E(y D X XDy) + 2E[y L X XDy θy XDy] R(Ly; β, σ 2 ). Thus Ly is the BLUE of β under the balanced loss (1.2). Necessity: If there exists a matrix D 0 such that E(D 0 y) = 0 and E(y L X XD 0 y θy XD 0 y) 0, then we denote k = E(y L X XD 0 y θy XD 0 y) and assume k < 0 (if k > 0, we can use D 0 in place of D 0 ). Supposing G = L + td 0, t R 1, we have R(Gy; β, σ 2 ) = E[θ(y XGy) (y XGy) + (1 θ)(gy β) X X(Gy β)] = R(Ly; β, σ 2 ) + t 2 E(y D 0X XD 0 y) + 2tE[y L X XD 0 y θy XD 0 y]. Note that t 2 E(y D 0 X XD 0 y) + 2tE[y L X XD 0 y θy XD 0 y] is a quadratic polynomial, and the coefficient of one-time term is a negative number, thus there exists a number t 0 such that t 2 0 E(y D 0 X XD 0 y)+2t 0 E[y L X XD 0 y θy XD 0 y] < 0. Therefore, denote G 0 = L + t 0 D 0, then G 0 y is a linear unbiased estimator of β and R(G 0 y; β, σ 2 ) < R(Ly; β, σ 2 ). This is a contradiction to that Ly is the BLUE of β. Theorem 4.1 Under the balanced loss function (1.2) and the model M 2, β BLUE 1 = β BLUE 2 for β if and only if N X V 2 T + 1 X = 0. Proof Necessity: According to Lemma 4.1, β BLUE 1 = β BLUE 2 if and only if E( β BLUE 1X XDy θy XDy) = 0 holds for all Dy U. This is equivalent to that E[y (θ(x X) 1 X + (1 θ)(x T + 1 X) 1 X T + 1 )X XMN X y θy XMN X y] = 0 holds for all M R p n, that is, tr(n X V 2 T + 1 X(X T + 1 X) 1 X XM) = 0 holds for all M R p n. Hence, N X V 2 T + 1 X = 0.

10 122 A^VÇÚO 1n ò Sufficiency: Let β BLUE 1 be the BLUE of β under the model M 1. If N X V 2 T 1 + X = 0, then N X V 2 T 1 + X(X T 1 + X) 1 X XM = 0, for all M R p n, that is, tr[t 1 + X(X T 1 + X) 1 X XMN X V 2 ] = 0 holds for all M R p n, which derives that E[y (θ(x X) 1 X + (1 θ)(x T + 1 X) 1 X T + 1 ) X XDy θy XDy] = 0 holds for all Dy U. Therefore, E( β BLUE 1X XDy θy XDy) = 0 holds for all Dy U. Thus β BLUE 1 is also the BLUE of β under the model M 2, i.e, β BLUE 1 = β BLUE 2. In the following, we will give an example to illustrate the use of the above result. such that Example 1 while the true model specifies that Let us follow Xu et al. (2011) to suppose that the adopted model is M 3 : y = 1 n β + ε 1, E(ε 1 ) = 0, Cov (ε 1 ) = σ 2 I n, M 4 : y = 1 n β + ε 2, E(ε 2 ) = 0, Cov (ε 2 ) = σ 2 [(1 ρ)i n + ρ1 n 1 n]. It is obvious that β = (1/n)1 ny is the BLUE of β under the model M 3 from Corollary 2.1. It is easy to verify that N X V 2 T + 1 X = 0 for X = 1 n, V 1 = I n and V 2 = (1 ρ)i n + σ 2 ρ1 n 1 n. Therefore, β is also the BLUE of β under the model M 4 according to Theorem Concluding Remarks In this paper, we provide the necessary and sufficient conditions for linear estimators to be the best linear unbiased estimator under Zellner s balanced loss function and the general Gauss-Markov model, and obtain the unique best linear unbiased estimator which is a balance between the least squares estimator and the optimal estimator under quadratic loss. The loss robustness of BLUE with respect to relative losses and relative saving losses are discussed, respectively. Furthermore, the robustness of BLUE on the mis-specification of the covariance matrix is also considered. It is interesting to observe that the best linear estimator is a balance between the least squares estimator and the best linear unbiased estimator under quadratic loss. Moreover, the weights assigned to the goodness of fit of

11 1Ï?m $±: ²ï e Gauss-Markov. 8Xê `O 123 model and the precision of estimation are consistent to the weights assigned to their corresponding optimal estimators. This makes us do further study in the future about the problem how to choose the proper weight when making use of the balanced loss function. References [1] Rao, C.R., Linear Statistical Inference and Its Applications, Wiley, New York, [2] Rao, C.R., Some recent results in linear estimation, Sankhyā: The Indian Journal of Statistics, Series B, 34(4)(1972), [3] Albert, A., The Gauss-Markov theorem for regression models with possibly singular covariances, SIAM Journal on Applied Mathematics, 24(2)(1973), [4] Zellner, A., Bayesian and non-bayesian estimation using balanced loss functions, In: Gupta, S.S. and Berger, J.O. (Editors), Statistical Decision Theory and Related Topics V, Springer, New York, 1994, [5] Rodrigues, J. and Zellner, A., Weighted balanced loss function and estimation of the mean time to failure, Communications in Statistics - Theory and Methods, 23(12)(1994), [6] Giles, J.A., Giles, D.E.A. and Ohtani, K., The exact risks of some pre-test and stein-type regression estimators umder balanced loss, Communications in Statistics - Theory and Methods, 25(12)(1996), [7] Ohtani, K., Giles, D.E.A. and Giles, J.A., The exact risk performance of a pre-test estimator in a heteroskedastic linear regression model under the balanced loss function, Econometric Reviews, 16(1)(1997), [8] Ohtani, K., the exact risk of a weighted average estimator of the OLS and Stein-rule estimators in regression under balanced loss, Statistics and Risk Modeling, 16(1)(1998), [9] Ohtani, K., Inadmissibility of the Stein-rule estimator under the balanced loss function, Journal of Econometrics, 88(1)(1999), [10] Gruber, M.H.J., The efficiency of shrinkage estimators with respect to Zellner s balanced loss function, Communications in Statistics - Theory and Methods, 33(2)(2004), [11] Jozani, M.J., Marchand, E. and Parsian, A., On estimation with weighted balanced-type loss function, Statistics and Probability Letters, 76(8)(2006), [12] Bansal, A.K. and Aggarwal, P., Bayes prediction for a heteroscedastic regression superpopulation model using balanced loss function, Communications in Statistics - Theory and Methods, 36(8)(2007), [13] Bansal, A.K. and Aggarwal, P., Bayes prediction of the regression coefficient in a finite population using balanced loss function, Metron - International Journal of Statistics, 67(1)(2009), [14] Bansal, A.K. and Aggarwal, P., Bayes prediction for a stratified regression superpopulation model using balanced loss function, Communications in Statistics - Theory and Methods, 39(15)(2010), [15] Hu, G.K. and Peng, P., Admissibility for linear estimators of regression coefficient in a general GaussõMarkoff model under balanced loss function, Journal of Statistical Planning and Inference, 140(11)(2010),

12 124 A^VÇÚO 1n ò [16] Hu, G.K. and Peng, P., All admissible linear estimators of a regression coefficient under a balanced loss function, Journal of Multivariate Analysis, 102(8)(2011), [17] Wang, S.G., The Theory of Linear Model and Its Application, Anhui Edcation Press, (in chinese) [18] Yu, S.H. and He, C.Z., Comparison of general Gauss-Markoff models in estimable subspace, Acta Mathematicae Applicatae Sinica, 20(4)(1997), (in Chinese) [19] Xu, L.W., Lu, M. and Jiang, C.F., Optimal prediction in finite populations under matrix loss, Journal of Statistical Planning and Inference, 141(8)(2011), ²ï e Gauss-Markov. 8Xê `O?m $ ± (ÀunóŒÆnÆ, H, ) 3²ï e, ïä Gauss-Markov. 8Xê `O, Äk 5O Z 5à O 7 ^ ; Ùgy²²ï e Z 5à O3A?? Âe, ÊÏ OÚg e `O²ï;,?Ø `O'u ¼êÚ.½ è5, T `O3.ؽeäk è5 7 ^. ' c: `O, è5, ²ï, Gauss-Markov.. Æ a Ò: O212.4.