We open this section with some notations: Given a matrix $A$, the symbols $\mathscr{M}(A), A'$, $\mathrm{tr}(A)$, $rk(A)$ will stand for the range space, the transpose, the trace, the rank, respectively, of matrix $A$. The $n\times n$ identity matrix is denoted by $I_{n}$. For an $n\times n$ matrix $A$, $A>0$ means that $A$ is a symmetric positive definite matrix. $A\geq 0$ means that $A$ is a symmetric nonnegative definite matrix, $A\geq B(A\leq B)$ means that $A-B\geq 0(B-A\leq 0)$. $R^{m\times n}$ stands for the set composed of all $m\times n$ real matrices.

Let us consider finite population $\mathscr{P}=\{1, \cdots, N\}$ as the collection of a known number $N$ of identifiable units. Associated with the $i$th unit of $\mathscr{P}$, there are $p+1$ quantities: $y_{i}, x_{i1}, \cdots, x_{ip}$, where all but $y_{i}$ are known, $i=1, \cdots, N$. Denote $y=(y_{1}, \cdots, y_{N})'$ and $X=(X_{1}, \cdots, X_{N})'$, where $X_{i}=(x_{i1}, \cdots, x_{ip})'$, $i=1, \cdots, N$. We express the relationships among the variables by the linear model

$ \begin{equation} y=X\beta+\varepsilon, \end{equation} $

(1.1)

where $\beta$ is a $p\times 1$ unknown superparameter vector, $\varepsilon$ is an $N\times 1$ random vector with mean $0$ and covariance matrix $\sigma^{2} V$, $V> 0$ is a known matrix, but the parameter $\sigma^{2}> 0$ is unknown. If $\varepsilon$ is a random vector with multivariate normal distribution, the model (1.1) will be written as

$ \begin{equation} y=X\beta+\varepsilon, \varepsilon\sim N(0, \sigma^{2}V). \end{equation} $

(1.2)

Denote the finite population regression coefficient as $\beta_{N}=(X'V^{-1}X)^{-1}X'V^{-1}y$. In the literature, a lot of predictions for finite population regression coefficient have been produced. For example, Bolfarine and Zacks [1, 2] studied Bayes and minimax prediction under square error loss function. Bolfarine et al. [3] obtained the best linear unbiased prediction under the generalized prediction mean squared error. Yu and Xu [4] studied the admissibility of linear predictors under quadratic loss. Xu and Yu [5] obtained the linear admissible predictors under matrix loss. Recently, Bansal and Aggarwal [6-8] considered Bayes prediction of finite population regression coefficient of some superpopulation regression model with normal assumption using Zellner's [9] balanced loss function. However, there is not much literature on the linear admissible prediction of the finite population regression coefficient in the superpopulation models with and without the assumption that the underlying distribution is normal using Zellner's balanced loss function.

In order to predict the finite population regression coefficient $\beta_{N}$, let us select a sample $\mathbf{s}$ of size $n(\leq N)$ from $\mathscr{P}$ according to some specified sampling plan in order to obtain information on $\beta_{N}$. Let $\mathbf{r}=\mathscr{P}-\mathbf{s}$ be the unobserved part of $\mathscr{P}$. After the sample has been selected, we may reorder the elements of $y$ such that we have the corresponding partitions of $y$, $X$ and $V$, that is

$ \begin{equation*} y=\left(\begin{array}{c}y_{s}\\ y_{r}\end{array}\right), X=\left(\begin{array}{c}X_{s}\\ X_{r}\end{array}\right), V=\left(\begin{array}{cc}V_{s}&V_{sr}\\ V_{rs}&V_{r}\end{array}\right). \end{equation*} $

Following Bolfarine et al.[3], we can write the finite population regression coefficient $\beta_{N}$ as

$ \begin{equation*} \beta_{N}=Q_{s}y_{s}+Q_{r}y_{r}, \end{equation*} $

where

$ \begin{eqnarray*} &&Q_{s}=F^{-1}BC^{-1}, Q_{r}=F^{-1}DE^{-1}, \\ &&B=X_{s}'-X_{r}'V_{r}^{-1}V_{rs}, C=V_{s}-V_{sr}V_{r}^{-1}V_{rs}, \\ &&D=X_{r}'-X_{s}'V_{s}^{-1}V_{sr}, E=V_{r}-V_{rs}V_{s}^{-1}V_{sr} \end{eqnarray*} $

and

$ F=BC^{-1}X_{s}+DE^{-1}X_{r}. $

Consider the class of homogeneous linear predictors as $\mathit{£} = \{ L{y_s}:L \in {R^{p \times n}}\} .$ Let $\delta(y_{s})$ be a predictor of $\beta_{N}$, in this article, we use Zellner's balanced loss function

$ \begin{equation} L(\delta(y_{s}), \beta_{N})=\theta (y_{s}-X_{s}\delta(y_{s}))'(y_{s}-X_{s}\delta(y_{s}))+(1-\theta)(\delta(y_{s})-\beta_{N})'X_{s}'X_{s}(\delta(y_{s})-\beta_{N}). \end{equation} $

(1.3)

Here $\theta\in [0, 1]$ is a weight coefficient. Zellner's balanced loss function takes both precision of estimation and goodness of fit of model into account, so it is a more comprehensive and reasonable standard than quadratic loss and residual sum of square. Moreover, we know that the balanced loss function is more sensitive than the quadratic loss function, which means that if a prediction is admissible under the balanced loss function, it is also admissible under the quadratic loss function. Therefore, the study about the admissible prediction under the balanced loss function is significant. Denote the corresponding risk function as $R(\delta(y_{s}), \beta_{N})=E(L(\delta(y_{s}), \beta_{N})).$

In this paper, we discuss the admissibility of linear predictors in the class of homogeneous linear predictors and in the class of all predictors, respectively.

The rest of this paper is organized as follows: Necessary and sufficient conditions for homogeneous linear predictors of $\beta_{N}$ to be admissible in $\mathit{£}$ under model (1.1) and loss function (1.3) are placed in Section 2. In Section 3, we give the sufficient conditions for homogeneous linear predictors to be admissible in the class of all predictors under model (1.2) and loss function (1.3). Concluding remarks are given in Section 4.

In this section, we give necessary and sufficient conditions for homogeneous linear predictors to be admissible in $\mathit{£}$ under the model (1.1) and the balanced loss function (1.3). First, we give a definition for admissibility.

Definition 2.1  The predictor $\delta_{1}(y_{s})$ is called as good as $\delta_{2}(y_{s})$ if and only if $R(\delta_{1}(y_{s}), \beta_{N})\leq R(\delta_{2}(y_{s}), \beta_{N})$ for all $\beta\in R^{p}$ and $\sigma^{2}>0$, and $\delta_{1}(y_{s})$ is called better than $\delta_{2}(y_{s})$ iff $\delta_{1}(y_{s})$ is as good as $\delta_{2}(y_{s})$ and $R(\delta_{1}(y_{s}), \beta_{N})\neq R(\delta_{2}(y_{s}), \beta_{N})$ at some $\beta_{0}\in R^{p}$ and $\sigma^{2}_{0}>0$. Let $\mathscr{L}$ be a class of predictors, then a predictor $\delta(y_{s})$ is said to be admissible for $\beta_{N}$ in $\mathscr{L}$ iff $\delta(y_{s})\in \mathscr{L}$ and there exists no predictor in $\mathscr{L}$ which is better than $\delta(y_{s})$.

Lemma 2.1 (Wu [10]) Consider the following model

$ \begin{equation} y_{s}=X_{s}\beta+\varepsilon_{s}, \end{equation} $

(2.1)

where $\varepsilon_{s}$ is a $n\times 1$ unobservable random vector with $E(\varepsilon_{s})=0, {\rm Cov}(\varepsilon_{s})=\sigma^{2}V_{s}$, $X_{s}$ and $V_{s}$ are known $n\times p$ and $n\times n$ matrices, respectively. Whereas $\beta\in R^{p}$ and $\sigma^{2}> 0$ are unknown parameters. If $S\beta$ is a linearly estimable variable under the model (2.1), then under the loss function $(d-S\beta)'(d-S\beta)$, $Ly_{s}$ is an admissible estimator of $S\beta$ in $\mathit{£}$ if and only if

(1) $L=LX_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}X_{s}'V_{s}^{-1}$,

(2) $LX_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}S'-LX_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}X_{s}'L'\geq 0$.

Theorem 2.1  Under model (1.1), $Ly_{s}$ is an admissible predictor of $\beta_{N}$ in $\mathit{£}$ under the balanced loss function (1.3) if and only if

(1)  $L=LX_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}X_{s}'V_{s}^{-1}+\theta(X_{s}'X_{s})^{-1}X_{s}'-\theta(X_{s}'V_{s}^{-1}X_{s})^{-1}X_{s}'V_{s}^{-1}$,

(2)  $(L-\theta(X_{s}'X_{s})^{-1}X_{s}'-(1-\theta)(Q_{s}+Q_{r}V_{rs}V_{s}^{-1}))X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}(I_{p}-X_{s}'L')\geq 0$.

Proof  By direct operation, we have

$ \;\;\;\;R(Ly_{s}, \beta_{N})\\ =\;E[\theta(y_{s}-X_{s}Ly_{s})'(y_{s}-X_{s}Ly_{s})\\+(1-\theta)(Ly_{s}-Q_{s}y_{s}-Q_{r}y_{r}) 'X_{s}'X_{s}(Ly_{s}-Q_{s}y_{s}-Q_{r}y_{r})]\\ =\;E[(\tilde{L}y_{s}-\tilde{S}\beta)'(\tilde{L}y_{s}-\tilde{S}\beta)]\\ \;\;\;\;+\sigma^{2}\mathrm{tr}[\theta V_{s}+(1-\theta)Q_{s}'X_{s}'X_{s}Q_{s}V_{s}+2(1-\theta)Q_{s}'X_{s}'X_{s}Q_{r}V_{rs}+(1-\theta)Q_{r}'X_{s}'X_{s}Q_{r}V_{r}]\\ \;\;\;\;-\sigma^{2}\mathrm{tr}[H(\theta H^{-1}X_{s}'+(1-\theta)(Q_{s}+Q_{r}V_{rs}V_{s}^{-1}))V_{s}(\theta H^{-1}X_{s}'\\\;\;\;\;+(1-\theta)(Q_{s}+Q_{r}V_{rs}V_{s}^{-1}))'], $

where

$ \begin{eqnarray*}&& H=X_{s}'X_{s}, \tilde{L}=H^{\frac{1}{2}}(L-\theta H^{-1}X_{s}'-(1-\theta)(Q_{s}+Q_{r}V_{rs}V_{s}^{-1})), \\ && \tilde{S}=(1-\theta)H^{\frac{1}{2}}(I_{p}-Q_{s}X_{s}-Q_{r}V_{rs}V_{s}^{-1}X_{s}).\end{eqnarray*} $

In order to prove that $Ly_{s}$ is an admissible prediction of $\beta_{N}$ in $\mathit{£}$ under balanced loss function (1.3), we need only to show that $\tilde{L}y_{s}$ is an admissible estimator for $\tilde{S}\beta$ under model (2.1) and loss function $(d-\tilde{S}\beta)'(d-\tilde{S}\beta)$ in $\mathit{£}$. It follows by Lemma 2.1 that $Ly_{s}$ is an admissible predictor of $\beta_{N}$ in $\mathit{£}$ under balanced loss function (1.3) if and only if

(1)  $\tilde{L}=\tilde{L}X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}X_{s}'V_{s}^{-1}$,

(2) $\tilde{L}X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}\tilde{S}'-\tilde{L}X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}X_{s}'\tilde{L}'\geq 0$.

On the basis of this, we can obtain the result by direct operation.

It is easy to obtain the following corollary by this theorem.

Corollary 2.1  Under model (1.1) and the loss function (1.3), $L_{1}y_{s}$ is an admissible predictor of $\beta_{N}$ in $\mathit{£}$, where $L_{1}=\theta (X_{s}'X_{s})^{-1}X_{s}'+(1-\theta)(X_{s}'V_{s}^{-1}X_{s})^{-1}X_{s}'V_{s}^{-1}$.

If $(y_{s}-X_{s}\delta(y_{s}))'(y_{s}-X_{s}\delta(y_{s}))$ is also considered to be a kind of loss function, then $(X_{s}'X_{s})^{-1}X_{s}'y_{s}$ is the best linear unbiased prediction in $\mathit{£}$. This corollary illustrates that the linear admissible prediction under loss (1.3) are convex combination between the linear admissible predictions under loss $(y_{s}-X_{s}\delta(y_{s}))'(y_{s}-X_{s}\delta(y_{s}))$ and the linear admissible predictions under loss $(\delta(y_{s})-\beta_{N})'X_{s}'X_{s}(\delta(y_{s})-\beta_{N})$. 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 admissible predictions. It is clear to illustrate the use of the balanced loss function (1.3).

In section 2, we have given the necessary and sufficient conditions for a homogeneous linear predictor to be admissible in the class of homogeneous linear predictors. It is interesting to discuss the problem whether the admissible predictor is also admissible in the class of all predictors. In this section, we will answer this problem under model (1.2) and the loss function (1.3). In the following, we first give some lemmas.

Lemma 3.1 (Wu [11]) Consider the following model

$ \begin{equation} \tilde{y}_{s}=\tilde{X}_{s}\beta+\tilde{\varepsilon}, \tilde{\varepsilon}\sim N(0, \sigma^{2}I_{n}), \end{equation} $

(3.1)

where $\tilde{y_{s}}\in R^{n}$, $\tilde{X}_{s}\in R^{n\times p}$. Let $L$ and $S$ be known $p\times n$ matrices. If $L$ satisfies the following conditions:

(1)  $L=L\tilde{X}_{s}(\tilde{X}_{s}'\tilde{X}_{s})^{-1}\tilde{X}_{s}', $

(2) $ L\tilde{X}_{s}(\tilde{X}_{s}'\tilde{X}_{s})^{-1}\tilde{X}_{s}'L'\leq L\tilde{X}_{s}(\tilde{X}_{s}'\tilde{X}_{s})^{-1}\tilde{X}_{s}'S', $

(3)  $rk(L\tilde{X}_{s}(\tilde{X}_{s}'\tilde{X}_{s})^{-1}\tilde{X}_{s}'(S-L)')\geq rk(L)-2.$

Then an estimator $L\tilde{y}_{s}$ of $S\tilde{X}_{s}\beta$ is admissible in the class of all estimators under loss function $(d-S\tilde{X}_{s}\beta)'(d-S\tilde{X}_{s}\beta)$.

Lemma 3.2  (Rao [12]) Let $L$ and $S$ be $m\times n$ matrices. Then $LS'$ is symmetric and $LL'\leq LS'$ if and only if there exists an $n\times n$ symmetric matrix $M\geq 0$ such that $L=SM$, $rk(M)=rk(L)$ and the eigenvalues of $M$ are in the closed interval $[0, 1]$.

Lemma 3.3  (Wu [11]) Let $L$ and $S$ be $m\times n$ matrices. Then the following two statements are equivalent.

(1) $LS'$ is symmetric, $LL'\leq LS'$ and $rk(LS'-LL')\geq rk(L)-2$,

(2) There exists an $n\times n$ symmetric matrix $M\geq 0$ such that $L=SM$, $rk(M)=rk(L)$, the eigenvalues of $M$ are in $[0, 1]$ and at most two of them are equal to one.

Lemma 3.4 (Rao [12]) Let $h(y)$ be an admissible estimator of $g(\gamma)$ under $(d-g(\gamma))'(d-g(\gamma))$. Then for every constant matrix $K$, $Kh(y)$ is an admissible estimator of $Kg(\gamma)$ under $(d_{1}-Kg(\gamma))'(d_{1}-Kg(\gamma)).$

Theorem 3.1  Under the model (1.2) and the loss function (1.3), a predictor $Ly_{s}$ of $\beta_{N}$ is admissible in the class of all predictors if $L$ satisfied the following conditions:

(1)  $\tilde{L}=\tilde{L}X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}X_{s}'V_{s}^{-1}, $

(2) $ \tilde{L}X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}X_{s}'\tilde{L}'\leq \tilde{L}X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}\tilde{S}', $

(3) $ rk(\tilde{L}X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}(\tilde{S}-\tilde{L}X_{s})')\geq, rk(\tilde{L})-2, $

where

$ \begin{eqnarray*}&& \tilde{L}=H^{\frac{1}{2}}(L-\theta H^{-1}X_{s}'-(1-\theta)(Q_{s}+Q_{r}V_{rs}V_{s}^{-1})), \\ && \tilde{S}=(1-\theta)H^{\frac{1}{2}}(I_{p}-Q_{s}X_{s}-Q_{r}V_{rs}V_{s}^{-1}X_{s}).\end{eqnarray*} $

Proof  According to the proof of Theorem 2.1, we have

$ \begin{eqnarray*} &&R(Ly_{s}, \beta_{N})\\ &=&E[(\tilde{L}y_{s}-\tilde{S}\beta)'(\tilde{L}y_{s}-\tilde{S}\beta)]+C^{2}\\ &=&E[(\tilde{L}_{1}\tilde{y}_{s}-\tilde{S}_{1}\tilde{X}_{s}\beta)'(\tilde{L}_{1}\tilde{y}_{s} -\tilde{S}_{1}\tilde{X}_{s}\beta)]+C^{2}, \end{eqnarray*} $

where

$ \begin{eqnarray*}&& \tilde{L}_{1}=\tilde{L}V_{s}^{\frac{1}{2}}, \tilde{y}_{s}=V_{s}^{-\frac{1}{2}}y_{s}, \tilde{X}_{s}=V_{s}^{-\frac{1}{2}}X_{s}, \\ && \tilde{S_{1}}=(1-\theta)H^{\frac{1}{2}}((X_{s}'X_{s})^{-1}X_{s}'-Q_{s}-Q_{r}V_{rs}V_{s}^{-1})V_{s}^{\frac{1}{2}}\end{eqnarray*} $

and

$ C^{2}\;=\;\sigma^{2}\mathrm{tr}[\theta V_{s}+(1-\theta)Q_{s}'X_{s}'X_{s}Q_{s}V_{s}+2(1-\theta)Q_{s}'X_{s}'X_{s}Q_{r}V_{rs}+(1-\theta)Q_{r}'X_{s}'X_{s}Q_{r}V_{r} \\ \;\;\;\;-H(\theta H^{-1}X_{s}'+(1-\theta)(Q_{s} +Q_{r}V_{rs}V_{s}^{-1}))V_{s}(\theta H^{-1}X_{s}'+(1-\theta)(Q_{s}\\\;\;\;\;+Q_{r}V_{rs}V_{s}^{-1}))']. $

Therefore, to prove that $Ly_{s}$ is an admissible predictor of $\beta_{N}$ in the class of all predictors, we need only to show that $\tilde{L}_{1}\tilde{y}_{s}$ is an admissible estimator of $\tilde{S}_{1}\tilde{X}_{s}\beta$ in the class of all estimators under model (3.1) and the loss function $(d-\tilde{S}_{1}\tilde{X}_{s}\beta)'(d-\tilde{S}_{1}\tilde{X}_{s}\beta)$. By Lemma 3.1, we obtain the result.

Corollary 3.1  Under model (1.2) and the loss function (1.3), $L_{1}y_{s}$ is admissible in the class of all predictors, where

$ L_{1}=\theta (X_{s}'X_{s})^{-1}X_{s}'+(1-\theta)(X_{s}'V_{s}^{-1}X_{s})^{-1}X_{s}'V_{s}^{-1}. $

The proof of this corollary is omitted here since it is easy to verify that $L_{1}y_{s}$ satisfies the conditions of Theorem 3.1.

Theorem 3.2  Let $\mathscr{M}(X_{s}'\tilde{L}')\subset \mathscr{M}(\tilde{S}')$ and $Ly_{s}$ be an admissible predictor of $\beta_{N}$ in the class of all predictors under the model (1.2) and the loss function (1.3). Then $L$ satisfies the following conditions:

(1)  $\tilde{L}=\tilde{L}X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}X_{s}'V_{s}^{-1}, $

(2) $ \tilde{L}X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}X_{s}'\tilde{L}'\leq \tilde{L}X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}\tilde{S}', $

(3) $ rk(\tilde{L}X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}(\tilde{S}-\tilde{L}X_{s})')\geq rk(\tilde{L})-2, $

where

$ \begin{eqnarray*}&& \tilde{L}=H^{\frac{1}{2}}(L-\theta H^{-1}X_{s}'-(1-\theta)(Q_{s}+Q_{r}V_{rs}V_{s}^{-1})), \\ && \tilde{S}=(1-\theta)H^{\frac{1}{2}}(I_{p}-Q_{s}X_{s}-Q_{r}V_{rs}V_{s}^{-1}X_{s}).\end{eqnarray*} $

Proof  Take an $n\times p$ matrix $\tilde{P}$ such that $\mathscr{M}(\tilde{P})=\mathscr{M}(\tilde{X}_{s})$ and $\tilde{P}'\tilde{P}=I_{p}$, where $p=rk(\tilde{X}_{s})$. Then $\tilde{X}_{s}(\tilde{X}_{s}'\tilde{X}_{s})^{-1}\tilde{X}_{s}'=\tilde{P}\tilde{P}'$. If $Ly_{s}\in \mathit{£}$ is an admissible predictor of $\beta_{N}$ in the class of all predictors under the model (1.2) and the loss function (1.3), then $\tilde{L}_{1}\tilde{y}_{s}$ is an admissible estimator of $\tilde{S}_{1}\tilde{X}_{s}\beta$ in the class of all estimators under the model (3.1) and the loss function $(d-\tilde{S}_{1}\tilde{X}_{s}\beta)'(d-\tilde{S}_{1}\tilde{X}_{s}\beta)$ according to the proof of Theorem 3.1. This shows that conditions (1) and (2) of this theorem hold by Lemma 2.1. Therefore, we will show that (3) holds using (1) and (2). Suppose, to the contrary, that (3) does not hold, i.e., $rk(\tilde{L}X_{s}(X_{s}'V_{s}^{-1}X_{s})^{-1}(\tilde{S}-\tilde{L}X_{s})')< rk(\tilde{L})-2$, which is equivalent to

$ rk(\tilde{L}_{1}\tilde{X}_{s}(\tilde{X}_{s}'\tilde{X}_{s})^{-1}\tilde{X}_{s}'(\tilde{S}_{1}-\tilde{L}_{1})')< rk(\tilde{L}_{1})-2. $

By equation $rk(\tilde{L}_{1})=rk(\tilde{L}_{1}\tilde{X}_{s})=rk(\tilde{L}_{1}\tilde{P})$, Lemmas 3.2, 3.3 and condition $\tilde{L}_{1}\tilde{P}\tilde{P}'\tilde{L}\leq \tilde{L}_{1}'\tilde{P}\tilde{P}'\tilde{S}_{1}'$, there exists a $p\times p$ symmetric matrix $M\geq 0$ such that $\tilde{L}_{1}\tilde{P}=\tilde{S}_{1}\tilde{P}M$, $rk(M)=rk(\tilde{L}_{1})$ and the eigenvalues of $M$ are in $[0, 1]$ and at least three eigenvalues are equal to 1. By the spectral decomposition of $\tilde{P}'\tilde{S}_{1}'\tilde{S}_{1}\tilde{P}$, we write $\tilde{P}'\tilde{S}_{1}'\tilde{S}_{1}\tilde{P}=\Gamma\tilde{\Delta}\Gamma'.$ Here $\Gamma$ is an orthogonal matrix of order $p$, $\tilde{\Delta}={\rm diag}(\tau_{1}, \cdots, \tau_{q}, 0\cdots, 0)$ and $\Delta={\rm diag}(\tau_{1}, \cdots, \tau_{q})>0$ with $q=rk(\tilde{S}_{1}\tilde{P})=rk(\tilde{S}_{1}\tilde{X}_{s})$. Since $\mathscr{M}(\tilde{X}_{s}'\tilde{L}_{1}')\subset \mathscr{M}(\tilde{X}_{s}'\tilde{S}_{1}')$ if and only if $\mathscr{M}(\tilde{P}'\tilde{L}_{1}')\subset \mathscr{M}(\tilde{P}'\tilde{S}_{1}')$, it follows from the definition of $M$ and $\mathscr{M}(X_{s}'\tilde{L}')\subset \mathscr{M}(\tilde{S}')$ that

$ \mathscr{M}(M)=\mathscr{M}(\tilde{P}'\tilde{L}_{1}')\subset \mathscr{M}(\tilde{P}'\tilde{S}_{1}')=\mathscr{M}(\tilde{P}'\tilde{S}_{1}'\tilde{S}_{1}\tilde{P}). $

Moreover, $\mathscr{M}(\Gamma'M\Gamma)\subset \mathscr{M}(\tilde{\Delta})$. This derives that $\Gamma' M\Gamma=\left( \begin{array}{cc} M_{1}&0\\ 0&0 \end{array}\right)$, where $M_{1}\geq 0$ is a symmetric matrix of order $q$ and has the same nonzero eigenvalues as those of $M$. Take a $q\times q$ orthogonal matrix $N_{1}$ such that $N'_{1}M_{1}N_{1}=\left( \begin{array}{cc} I_{t}&0\\ 0&U \end{array}\right)$, where $t\geq 3$ and $U=diag(\omega_{1}, \cdots, \omega_{q-t})\geq 0$. Then $N=\left( \begin{array}{cc} N_{1}&0\\ 0&I_{p-q} \end{array}\right)$ is an orthogonal matrix of order $p$ and

$ \begin{equation*} N'\Gamma'M\Gamma N=\left( \begin{array}{ccc} I_{t}&0&0\\ 0&U&0\\ 0&0&0 \end{array}\right)\triangleq G. \end{equation*} $

We here write

$ \begin{equation} z=N'\Gamma'\tilde{P}'\tilde{y}_{s}, \gamma=N'\Gamma'\tilde{P}'\tilde{X}_{s}\beta \end{equation} $

(3.2)

and $(n-p)\hat{\sigma}^{2}=\tilde{y}'_{s}(I_{n}-\tilde{X}_{s}(\tilde{X}'_{s}\tilde{X}_{s})^{-1}\tilde{X}'_{s})\tilde{y}_{s}$. Then eq. (3.2) implies that

$ \begin{equation} z\sim N_{p}(\gamma, \sigma^{2}I_{p}), (n-p)\sigma^{-2}\hat{\sigma}^{2}\sim \chi^{2}_{n-p}, \end{equation} $

(3.3)

and $z, \hat{\sigma}^{2}$ are mutually independent. We also have

$ \begin{equation} \tilde{L}_{1}\tilde{y}_{s}=\tilde{S}_{1}\tilde{P}\Gamma NGz, \tilde{S}_{1}\tilde{X}_{s}\beta=\tilde{S}_{1}\tilde{P}\Gamma N\gamma. \end{equation} $

(3.4)

By $\tilde{S}_{1}\tilde{P}\Gamma(\Gamma'\tilde{P}'\tilde{S}_{1}'\tilde{S}_{1}\tilde{P}\Gamma)^{-}\Gamma'\tilde{P}'\tilde{S}_{1}'\tilde{S}_{1}\tilde{P}\Gamma =\tilde{S}_{1}\tilde{P}\Gamma$, Lemma 3.4 and eq. (3.4), $\tilde{L}_{1}\tilde{y}_{s}$ is an admissible estimator of $\tilde{S}_{1}\tilde{X}_{s}\beta$ under loss $(d-\tilde{S}_{1}\tilde{X}_{s}\beta)'(d-\tilde{S}_{1}\tilde{X}_{s}\beta)$ if and only if $\Gamma'\tilde{P}'\tilde{S}_{1}'\tilde{S}_{1}\tilde{P}\Gamma NGz$ is an admissible estimator of $g(\gamma)$ under the loss function $(d-g(\gamma))'(d-g(\gamma))$, where

$ g(\gamma)=\Gamma'\tilde{P}'\tilde{S}_{1}'\tilde{S}_{1}\tilde{P}\Gamma N\gamma. $

Since $N'\left( \begin{array}{cc} \Delta^{-1}&0\\ 0&I_{p-q} \end{array}\right)\Gamma'\tilde{P}'\tilde{S}_{1}'\tilde{S}_{1}\tilde{P}\Gamma NGz=Gz~~{\rm and} ~~N'\left( \begin{array}{cc} \Delta^{-1}&0\\ 0&I_{p-q} \end{array}\right)\Gamma'\tilde{P}'\tilde{S}_{1}'\tilde{S}_{1}\tilde{P}\Gamma N\gamma=\tilde{I_{q}}\gamma, $ where $\tilde{I_{q}}=\left( \begin{array}{cc} I_{q}&0\\ 0&0 \end{array}\right)$, $\tilde{L}_{1}\tilde{y}_{s}$ is an admissible estimator of $\tilde{S}_{1}\tilde{X}_{s}\beta$ under loss $(d-\tilde{S}_{1}\tilde{X}_{s}\beta)'(d-\tilde{S}_{1}\tilde{X}_{s}\beta)$ if and only if $Gz$ is an admissible estimator of $\tilde{I_{q}}\gamma$ under the loss function $(d-\tilde{I_{q}}\gamma)'(d-\tilde{I_{q}}\gamma)$ by Lemma 3.4. Partition $z$ and $\gamma$ as

$ \begin{equation*} z=\left( \begin{array}{c} z_{(1)} \\ z_{(2)} \\ z_{(3)} \end{array}\right)~ \begin{array}{c} t\times 1 \\ (q-t)\times 1 \\ (p-q)\times 1 \end{array}, \quad \gamma=\left( \begin{array}{c} \gamma_{(1)} \\ \gamma_{(2)} \\ \gamma_{(3)} \end{array}\right)~ \begin{array}{c} t\times 1 \\ (q-t)\times 1 \\ (p-q)\times 1 \end{array}. \end{equation*} $

Then the loss of $Gz$ is expressed as

$ \begin{eqnarray*}&& (Gz-\tilde{I_{q}}\gamma)'(Gz-\tilde{I_{q}}\gamma)\\ &=&(z_{(1)}-\gamma_{(1)})'(z_{(1)}-\gamma_{(1)}) +(Uz_{(2)}-\gamma_{(2)})'(Uz_{(2)}-\gamma_{(2)}).\end{eqnarray*} $

Hence, to verify this theorem, we need only to show that $z_{(1)}$ is an inadmissible estimator of $\gamma_{(1)}$ under the loss function $(d-\gamma_{(1)})'(d-\gamma_{(1)})$. By eq. (3.3), $z_{(1)}\sim N_{t}(\gamma_{(1)}, \sigma^{2}I_{t})$ with $t\geq 3$. Take

$ w=[1-2c(n-p)(n-p+2)^{-1}\hat{\sigma}^{2}(z'_{(1)}z_{(1)})^{-1}]z_{(1)} $

with a constant $c$. Using integration by parts, we have

$ \begin{eqnarray*} &&E(z_{(1)}-\gamma_{(1)})'(z_{(1)}-\gamma_{(1)})-E(w-\gamma_{(1)})'(w-\gamma_{(1)})\\ &=&\frac{4c(n-p)}{(n-p+2)}E[\frac{(z_{(1)}-\gamma_{(1)})'z_{(1)}\hat{\sigma}^{2}}{z'_{(1)}z_{(1)}}- \frac{c(n-p)\hat{\sigma}^{4}}{(n-p+2)z'_{(1)}z_{(1)}}]\\ &=&\frac{4c(n-p)(t-2-c)\sigma^{4}}{n-p+2}E(\frac{1}{z'_{(1)}z_{(1)}})>0, \end{eqnarray*} $

if $c$ is specified as one satisfying $0<c<t-2$. This proves that $z_{(1)}$ is inadmissible.

In this paper, necessary and sufficient conditions are given for homogeneous linear predictors to be admissible in the class of homogeneous linear predictors under the linear model (1.1). Sufficient conditions are also given for homogeneous linear predictors to be admissible in the class of all predictors under the linear model (1.2). They are proved to be necessary under additional conditions. However, it is also interesting to study the minimaxity of homogeneous linear predictors of the finite population regression coefficient under a balanced loss function. This will be studied in the other paper.