Admissible linear estimators in the general Gauss–Markov model under generalized extended balanced loss function

Abstract

This paper proposes a new generalized extended balanced loss function (GEBLF). Admissibility of linear estimators is characterized in the General Gauss–Markov model with respect to GEBLF. The sufficient and necessary conditions for linear estimators to be admissible with a dispersion matrix possibly singular among the set of linear estimators are obtained. It is stated that the results obtained under special conditions lead to the results known in the literature.

Appendices

Appendix 1: The proof of Eq. (10) in Theorem 1

Proof

If $\beta_{*} = Fy \in {\mathcal{F}}^{h}$, using some properties of trace and Lemma 1, then risk function of GEBLF is given by.

$$ \begin{aligned} R\left( {Fy;\beta ,\sigma^{2} } \right) & = \sigma^{2} tr\left[ \lambda_{1} \left( {I_{n} - XF} \right)^{\prime } T^{ + } \left( {I_{n} - XF} \right)V \right. \\ &\quad \left. + \lambda_{2} (F^{\prime}\tilde{S}FV) + \lambda_{3} \left( {\left( {XF - I_{n} } \right)^{\prime } T^{ + } XFV} \right) \right] \hfill \\ & \quad + \beta^{\prime}\left( {FX - I_{p} } \right)^{\prime } \left[ {\lambda_{1} X^{\prime}T^{ + } X + \lambda_{2} \tilde{S} + \lambda_{3} X^{\prime}T^{ + } X} \right]\left( {FX - I_{p} } \right)\beta \,, \hfill \\ \end{aligned} $$

Let $B = \lambda_{1} X^{\prime}T^{ + } X + \lambda_{2} \tilde{S} + \lambda_{3} X^{\prime}T^{ + } X = \lambda_{2} \tilde{S} + (1 - \lambda_{2} )S_{T}$, where $S_{T} = X^{\prime}T^{ + } X$. In this case, we get

$$ \begin{array}{*{20}l} {R\left( {Fy;\beta ,\sigma ^{2} } \right) = \sigma ^{2} tr\left[ {F^{\prime}BF - \left( {1 + \lambda _{1} - \lambda _{2} } \right)T^{ + } XF} \right]V + \sigma ^{2} tr(\lambda _{1} T^{ + } V)} \hfill \\ {\qquad\;\;\; \quad \quad \quad \quad \; + \beta ^{\prime}\left( {FX - I_{p} } \right)^{\prime } B\left( {FX - I_{p} } \right)\beta } \hfill \\ \end{array} . $$

Since $0 \le \lambda_{1} ,\lambda_{2} \le 1$, we can find such a $w$ denoted by $w = {{\left( {1 + \lambda_{1} - \lambda_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {1 + \lambda_{1} - \lambda_{2} } \right)} 2}} \right. \kern-0pt} 2}$, where $0 \le w \le 1$. So, Eq. (10) can be rewritten as follows for $0 \le \lambda_{1} ,\lambda_{2} \le 1$:

$$ \begin{aligned} &R\left( {Fy;\beta ,\sigma^{2} } \right) = \sigma^{2} tr\left[ {\left( {F - wB^{ - 1} X^{\prime}T^{ + } } \right)^{\prime } B\left( {F - wB^{ - 1} X^{\prime}T^{ + } } \right)} \right]V \hfill \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \sigma^{2} tr(\lambda_{1} T^{ + } V - w^{2} T^{ + } XB^{ - 1} X^{\prime}T^{ + } ) + \beta^{\prime}\left( {FX - I_{p} } \right)^{\prime } B\left( {FX - I_{p} } \right)\beta \,. \hfill \\ \end{aligned} $$

(31)

If we take $F^{*} = B^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \tilde{F} = B^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \left( {F - wB^{ - 1} X^{\prime}T^{ + } } \right)$$C = B^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} (I_{p} - wB^{ - 1} S_{T} )$, $B = B^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} B^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}}$.

then (31) equal to

$$ \begin{array}{*{20}l} {R\left( {Fy;\beta ,\sigma ^{2} } \right) = \sigma ^{2} tr(F^{*} VF^{{*\prime }} ) + \sigma ^{2} tr(\lambda _{1} T^{ + } V - w^{2} T^{ + } XB^{{ - 1}} X^{\prime } T^{ + } )} \hfill \\ {\quad\; \qquad \quad \quad \quad\; \quad + \beta ^{\prime } \left\{ {F^{*} X - C} \right\}^{\prime } \left\{ {F^{*} X - C} \right\}\beta } \hfill \\ \end{array} . $$

Appendix 2: The proof of Eq. (21) in Corollary 2

$$ \begin{gathered} P_{XT} = X\left[ {X^{\prime}\left( {V + XX^{\prime}} \right)^{ - 1} X} \right]^{ - } X^{\prime}\left( {V + XX^{\prime}} \right)^{ - 1} \hfill \\ \,\,\,\,\,\,\,\,\, = X\left[ {S_{V} - S_{V} \left( {I + S_{V} } \right)^{ - 1} S_{V} } \right]^{ - } X^{\prime}\,\left[ {V^{ - 1} - V^{ - 1} X\left( {I + S_{V} } \right)^{ - 1} X^{\prime}V^{ - 1} } \right] \hfill \\ \,\,\,\,\,\,\,\,\, = XS_{V}^{ - } \left( {I + S_{V} } \right)\left( {I + S_{V} } \right)^{ - 1} X^{\prime}V^{ - 1} \hfill \\ \,\,\,\,\,\,\,\,\, = XS_{V}^{ - } X^{\prime}V^{ - 1} \,\, = P_{XV} \, \hfill \\ \end{gathered} $$

Appendix 3: The proof of Eq. (22) in Theorem 4

Let $B = \lambda_{2} \tilde{S} + (1 - \lambda_{2} )S_{T}$. Then

$$ \begin{array}{*{20}l} {R\left( {Fy + f;\beta ,\sigma^{2} } \right) = E\left[ {\lambda_{1} \left( {y - X(Fy + f)} \right)^{\prime } T^{ + } \left( {y - X(Fy + f)} \right)} \right]} \hfill \\ {\quad + E\left[ {\lambda_{2} \left( {Fy + f - \beta } \right)^{\prime } \tilde{S}\left( {Fy + f - \beta } \right) + \lambda_{3} \left( {X(Fy + f) - y} \right)^{\prime } T^{ + } X\left( {Fy + f - \beta } \right)} \right]} \hfill \\ {\quad = \sigma^{2} tr\left[ {\lambda_{1} \left( {I_{n} - XF} \right)^{\prime } T^{ + } \left( {I_{n} - XF} \right) + \lambda_{2} \left( {F^{\prime}\tilde{S}F} \right) + \lambda_{3} \left( {\left( {XF - I_{n} } \right)^{\prime } T^{ + } XF} \right)} \right]V} \hfill \\ {\quad + \left[ {\left( {FX - I_{p} } \right)\beta + f} \right]^{\prime } \left[ {\lambda_{1} S_{T} + \lambda_{2} \tilde{S} + \lambda_{3} S_{T} } \right]\left[ {\left( {FX - I_{p} } \right)\beta + f} \right]} \hfill \\ {\quad = \sigma^{2} tr\left[ {\lambda_{1} \left( {I_{n} - XF} \right)^{\prime } T^{ + } \left( {I_{n} - XF} \right) + \lambda_{2} \left( {F^{\prime}\tilde{S}F} \right) + \lambda_{3} \left( {\left( {XF - I_{n} } \right)^{\prime } T^{ + } XF} \right)} \right]V} \hfill \\ {\quad + \left[ {\left( {FX - I_{p} } \right)\beta + f} \right]^{\prime } B\left[ {\left( {FX - I_{p} } \right)\beta + f} \right].} \hfill \\ \end{array} $$