# 两线性随机效应模型未知参数函数预测量的等价性研究Equivalence Study of Unknown Parameter Function Predictors under Two Linear Random Effects Models

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Suppose that adding a new variable term to the general linear random-effect model becomes the overparameter linear random-effect model, then the statistical inference of the some unknown parameter of the two models is not necessarily the same. In order to solve this problem, this paper uses the solution of constrained quadratic matrix-valued function optimization problem to give the analytical expression of the best linear unbiased predictor/best linear unbiased estimator of the unknown parameter function under the overparameter linear random-effect model. The conditions for the equivalence of the best linear unbiased predictor/best linear unbiased estimator of unknown parameter functions of two models are obtained by using some algebra and matrix theory tools.

1. 引言

2. 模型与方法

${M}_{1}:\text{ }\text{ }\text{ }\text{ }\text{ }y=X\beta +\epsilon ,\text{ }\text{ }\beta =Z\alpha +\gamma$ (1)

${M}_{2}:\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }y=X\beta +F\zeta +\epsilon ,\text{ }\text{ }\beta =Z\alpha +\gamma$ (2)

$E\left[\begin{array}{l}\gamma \\ \epsilon \end{array}\right]=0$ , $Cov\left[\begin{array}{l}\gamma \\ \epsilon \end{array}\right]=\left[\begin{array}{cc}{\Sigma }_{11}& {\Sigma }_{12}\\ {\Sigma }_{21}& {\Sigma }_{22}\end{array}\right]=\Sigma$ (3)

$D\left(y\right)=D\left(X\gamma +\epsilon \right)=X{\Sigma }_{11}{X}^{\prime }+X{\Sigma }_{12}+{\Sigma }_{21}{X}^{\prime }+{\Sigma }_{22}=\stackrel{˜}{X}\Sigma {\stackrel{˜}{X}}^{\prime }$ (4)

$Cov\left(X\gamma ,y\right)=Cov\left(X\gamma ,X\gamma +\epsilon \right)=X{\Sigma }_{11}{X}^{\prime }+{\Sigma }_{12}=X\Sigma {\stackrel{˜}{X}}^{\prime }$ (5)

(6)

$\varphi =A\alpha +B\gamma +C\epsilon$ (7)

$f\left({L}_{u}\right)=\left({L}_{u}{C}_{1}+{D}_{1}\right)M{\left({L}_{u}{C}_{1}+{D}_{1}\right)}^{\prime }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{s}\text{.t}\text{ }\text{ }\text{ }\text{ }\text{ }{L}_{u}{A}_{1}={B}_{1}$

$f\left({L}_{u}\right)\ge f\left(L0\right)$

${L}_{0}\left[{A}_{1},{C}_{1}M{{C}^{\prime }}_{1}{A}_{1}^{\perp }\right]=\left[{B}_{1},-{D}_{1}M{{C}^{\prime }}_{1}{A}_{1}^{\perp }\right]$

${L}_{0}$ 的一般表达式及相应地 $f\left({L}_{0}\right)$$f\left({L}_{u}\right)-f\left({L}_{0}\right)$ 如下

${L}_{0}=\mathrm{arg}\underset{{L}_{u}{A}_{1}={B}_{1}}{\mathrm{min}}f\left({L}_{u}\right)=\left[{B}_{1},-{D}_{1}M{{C}^{\prime }}_{1}{A}_{1}^{\perp }\right]{\left[{A}_{1},{C}_{1}M{{C}^{\prime }}_{1}{A}_{1}^{\perp }\right]}^{+}+U{\left[{A}_{1},{C}_{1}M{{C}^{\prime }}_{1}\right]}^{\perp }$

$f\left({L}_{0}\right)=\underset{{L}_{u}{A}_{1}={B}_{1}}{\mathrm{min}}f\left({L}_{u}\right)=KM{K}^{\prime }-KM{C}_{1}T{{C}^{\prime }}_{1}M{K}^{\prime }$

$\begin{array}{c}f\left({L}_{u}\right)=f\left({L}_{0}\right)+\left({L}_{u}{C}_{1}+{D}_{1}\right)M{{C}^{\prime }}_{1}T{C}_{1}M{\left({L}_{u}{C}_{1}+{D}_{1}\right)}^{\prime }\\ =f\left({L}_{0}\right)+\left({L}_{u}{C}_{1}M{{C}^{\prime }}_{1}{A}_{1}^{\perp }+{D}_{1}M{{C}^{\prime }}_{1}{A}_{1}^{\perp }\right)T{\left({L}_{u}{C}_{1}M{{C}^{\prime }}_{1}{A}_{1}^{\perp }+{D}_{1}M{{C}^{\prime }}_{1}{A}_{1}^{\perp }\right)}^{\prime }\end{array}$

$r\left[{A}_{2},{B}_{2}\right]=r\left({A}_{2}\right)+r\left({E}_{{A}_{2}}{B}_{2}\right)=r\left({B}_{2}\right)+r\left({E}_{{B}_{2}}{A}_{2}\right)$ (8)

$r\left[\begin{array}{l}{A}_{2}\\ {C}_{2}\end{array}\right]=r\left({A}_{2}\right)+r\left({C}_{2}{F}_{{A}_{2}}\right)=r\left({C}_{2}\right)+r\left({A}_{2}{F}_{{C}_{2}}\right)$ (9)

$r\left[\begin{array}{cc}{A}_{2}& {B}_{2}\\ {C}_{2}& 0\end{array}\right]=r\left({B}_{2}\right)+r\left({C}_{2}\right)+r\left({E}_{{B}_{2}}{A}_{2}{F}_{{C}_{2}}\right)$ (10)

3. 过参数线性随机效应模型向量函数 $\varphi$ 的BLUP/BLUE及性质

$R\left({\stackrel{^}{X}}^{\prime }\right)\supseteq R\left({A}^{\prime }\right)$ (11)

$R\left({A}^{\prime }\right)$ 表示 ${A}^{\prime }$ 的列空间，

$E\left({L}_{1}y-\varphi \right)=0$$D\left({L}_{1}y-\varphi \right)=\mathrm{min}⇔{L}_{1}\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp }\right]=\left[A,N{\stackrel{^}{X}}^{\perp }\right]$ (12)

$BLU{P}_{{M}_{1}}\left(\varphi \right)={L}_{1}y=\left(\left[A,N{\stackrel{^}{X}}^{\perp }\right]{\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp }\right]}^{+}+{U}_{1}{\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp }\right]}^{\perp }\right)y$ (13)

$BLU{E}_{{M}_{1}}\left(A\alpha \right)=\left(\left[A,0\right]{\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp }\right]}^{+}+{U}_{1}{\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp }\right]}^{\perp }\right)y$ (14)

(a) $r\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp }\right]=r\left[\stackrel{^}{X},V\right]$$R\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp }\right]=R\left[\stackrel{^}{X},V\right]$$R\left(\stackrel{^}{X}\right)\cap R\left(V\stackrel{^}{X}\right)=\left\{\text{0}\right\}$

(b) ${L}_{1}$ 唯一的充要条件是 $R\left[\stackrel{^}{X},V\right]=n$

(c) $BLU{P}_{{M}_{1}}\left(\varphi \right)$ 以概率1唯一的充要条件是 $y\in R\left[\stackrel{^}{X},V\right]$，且模型(1)是相容的。

$\varphi ={A}^{*}{\alpha }^{*}+B\gamma +C\epsilon =\left[A,0\right]\left[\begin{array}{l}\alpha \\ \zeta \end{array}\right]+B\gamma +C\epsilon$ (15)

$\begin{array}{c}{L}_{2}y-\varphi ={L}_{2}XZ\alpha +{L}_{2}F\zeta +{L}_{2}X\gamma +{L}_{2}\epsilon -{A}^{*}{\alpha }^{*}-B\gamma -C\epsilon \\ =\left[{L}_{2}\stackrel{^}{X},{L}_{2}F\right]{\alpha }^{*}+{L}_{2}X\gamma +{L}_{2}\epsilon -{A}^{*}{\alpha }^{*}-B\gamma -C\epsilon \\ =\left(\left[{L}_{2}\stackrel{^}{X},{L}_{2}F\right]-{A}^{*}\right){\alpha }^{*}+\left({L}_{2}X-B\right)\gamma +\left({L}_{2}-C\right)\epsilon \\ =\left({L}_{2}W-{A}^{*}\right){\alpha }^{*}+\left({L}_{2}X-B\right)\gamma +\left({L}_{2}-C\right)\epsilon \end{array}$ (16)

$E\left({L}_{2}y-\varphi \right)=E\left[\left({L}_{2}W-{A}^{*}\right){\alpha }^{*}\right]=\left({L}_{2}W-{A}^{*}\right){\alpha }^{*}$ (17)

$\begin{array}{c}Cov\left({L}_{2}y-\varphi \right)=D\left[\left({L}_{2}X-B\right)\gamma +\left({L}_{2}-C\right)\epsilon \right]\\ =\left[\left({L}_{2}X-B\right),\left({L}_{2}-C\right)\right]\Sigma {\left[\left({L}_{2}X-B\right),\left({L}_{2}-C\right)\right]}^{\prime }\\ =\left({L}_{2}\left[X,{I}_{n}\right]-\left[B,C\right]\right)\Sigma {\left({L}_{2}\left[X,{I}_{n}\right]-\left[B,C\right]\right)}^{\prime }\\ =\left({L}_{2}\stackrel{˜}{X}-H\right)\Sigma {\left({L}_{2}\stackrel{˜}{X}-H\right)}^{\prime }:=f\left({L}_{2}\right)\end{array}$ (18)

$R\left({\left[\stackrel{^}{X},F\right]}^{\prime }\right)\supseteq R\left({\left[A,0\right]}^{\prime }\right)$ (19)

$E\left({L}_{2}y-\varphi \right)=0⇔\left[{L}_{2}\stackrel{^}{X},{L}_{2}F\right]{\alpha }^{*}-\left[A,0\right]{\alpha }^{*}$ 对所有的 ${\alpha }^{*}⇔{L}_{2}\left[\stackrel{^}{X},F\right]=\left[A,0\right]$

$E\left({L}_{2}y-\varphi \right)=0$$D\left({L}_{2}y-\varphi \right)=\mathrm{min}⇔{L}_{2}\left[W,V{W}^{\perp }\right]=\left[{A}^{*},N{W}^{\perp }\right]$ (20)

$BLU{P}_{{M}_{2}}\left(\varphi \right)={L}_{2}y=\left(\left[{A}^{*},N{W}^{\perp }\right]{\left[W,V{W}^{\perp }\right]}^{+}+{U}_{2}{\left[W,V{W}^{\perp }\right]}^{\perp }\right)y$ (21)

$BLU{E}_{{M}_{2}}\left({A}^{*}\alpha \right)=\left(\left[{A}^{*},0\right]{\left[W,V{W}^{\perp }\right]}^{+}+{U}_{2}{\left[W,V{W}^{\perp }\right]}^{\perp }\right)y$ (22)

(a) $BLU{P}_{{M}_{2}}\left(\varphi \right)$$BLU{P}_{{M}_{2}}\left(\varphi \right)$$\varphi$ 的协方差矩阵如下

$D\left[BLU{P}_{{M}_{2}}\left(\varphi \right)\right]=\left[{A}^{*},N{W}^{\perp }\right]{\left[W,V{W}^{\perp }\right]}^{+}V{\left(\left[A,N{W}^{\perp }\right]{\left[W,V{W}^{\perp }\right]}^{+}\right)}^{\prime }$ (23)

$Cov\left(BLU{P}_{{M}_{2}}\left(\varphi \right),\varphi \right)=\left[{A}^{*},N{W}^{\perp }\right]{\left[W,V{W}^{\perp }\right]}^{+}\stackrel{˜}{X}\Sigma {H}^{\prime }$ (24)

$D\left(\varphi \right)-D\left[BLU{P}_{{M}_{2}}\left(\varphi \right)\right]=H\Sigma {H}^{\prime }-\left[{A}^{*},N{W}^{\perp }\right]{\left[W,V{W}^{\perp }\right]}^{+}V{\left(\left[{A}^{*},N{W}^{\perp }\right]{\left[W,V{W}^{\perp }\right]}^{+}\right)}^{\prime }$ (25)

$D\left[\varphi -BLU{P}_{{M}_{2}}\left(\varphi \right)\right]=\left(\left[{A}^{*},N{W}^{\perp }\right]{\left[W,V{W}^{\perp }\right]}^{+}\stackrel{˜}{X}-H\right)\Sigma {\left(\left[{A}^{*},N{W}^{\perp }\right]{\left[W,V{W}^{\perp }\right]}^{+}\stackrel{˜}{X}-H\right)}^{\prime }$ (26)

(b) $\varphi$$BLUPs$ 能被分解为

$BLU{P}_{{M}_{2}}\left(\varphi \right)=BLU{P}_{{M}_{2}}\left(A\alpha \right)+BLU{P}_{{M}_{2}}\left(B\gamma \right)+BLU{P}_{{M}_{2}}\left(C\epsilon \right)$ (27)

(c) 对所有的 $T\in {ℝ}^{t×s}$ 满足 $BLU{P}_{{M}_{2}}\left(T\varphi \right)=TBLU{P}_{{M}_{2}}\left(\varphi \right)$

(d) 特别地

$BLU{P}_{{M}_{2}}\left(A\alpha \right)=\left(\left[{A}^{*},0\right]{\left[W,V{W}^{\perp }\right]}^{+}+{U}_{2}\left[W,V{W}^{\perp }\right]\right)y$ (28)

$D\left(BLU{P}_{{M}_{2}}\left(A\alpha \right)\right)=\left[{A}^{*},0\right]{\left[W,V{W}^{\perp }\right]}^{+}V{\left(\left[{A}^{*},0\right]{\left[W,V{W}^{\perp }\right]}^{+}\right)}^{\prime }$ (29)

4. 两模型下向量函数 $\varphi$ 的BLUP/BLUE的等价条件

$BLU{P}_{{M}_{1}}\left(\varphi \right)={L}_{1}y$$BLU{P}_{{M}_{2}}\left(\varphi \right)={L}_{2}y$ (30)

${L}_{1}=\left[A,N{\stackrel{^}{X}}^{\perp }\right]{\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp }\right]}^{+}+{U}_{1}{\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp }\right]}^{\perp }$ (31)

${L}_{2}=\left[{A}^{*},N{W}^{\perp }\right]{\left[W,V{W}^{\perp }\right]}^{+}+{U}_{2}{\left[W,V{W}^{\perp }\right]}^{\perp }$ (32)

(a) $BLU{P}_{{M}_{1}}\left(\varphi \right)=BLU{P}_{{M}_{2}}\left(\varphi \right)$

(b) $r\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\\ N& A& 0\end{array}\right]=r\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\end{array}\right]$

(c) $r\left[\begin{array}{cc}{F}^{\perp }V{\stackrel{^}{X}}^{\perp }& {F}^{\perp }\stackrel{^}{X}\\ N{\stackrel{^}{X}}^{\perp }& A\end{array}\right]=r\left[{F}^{\perp }V{\stackrel{^}{X}}^{\perp },{F}^{\perp }\stackrel{^}{X}\right]$

(d) $R\left({\left[\begin{array}{ccc}N& A& 0\end{array}\right]}^{\prime }\right)\subseteq R\left({\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\end{array}\right]}^{\prime }\right)$

(e) $R\left({\left[N{\stackrel{^}{X}}^{\perp },A\right]}^{\prime }\right)\subseteq R\left({\left[{F}^{\perp }V{\stackrel{^}{X}}^{\perp },{F}^{\perp }\stackrel{^}{X}\right]}^{\prime }\right)$

${L}_{0}\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp },W,V{W}^{\perp }\right]=\left[A,N{\stackrel{^}{X}}^{\perp },{A}^{*},N{W}^{\perp }\right]$

$r\left[\begin{array}{l}\begin{array}{cccc}\stackrel{^}{X}& V{\stackrel{^}{X}}^{\perp }& W& V{W}^{\perp }\end{array}\\ \begin{array}{cccc}A& N{\stackrel{^}{X}}^{\perp }& {A}^{*}& N{W}^{\perp }\end{array}\end{array}\right]=r\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp },W,V{W}^{\perp }\right]$

$\begin{array}{l}r\left[\begin{array}{cccc}\stackrel{^}{X}& V{\stackrel{^}{X}}^{\perp }& W& V{W}^{\perp }\\ A& N{\stackrel{^}{X}}^{\perp }& {A}^{*}& N{W}^{\perp }\end{array}\right]\\ =r\left[\begin{array}{ccccc}\stackrel{^}{X}& V{\stackrel{^}{X}}^{\perp }& \stackrel{^}{X}& F& V{W}^{\perp }\\ A& N{\stackrel{^}{X}}^{\perp }& A& 0& N{W}^{\perp }\end{array}\right]=r\left[\begin{array}{ccc}\stackrel{^}{X}& V{\stackrel{^}{X}}^{\perp }& F\\ A& N{\stackrel{^}{X}}^{\perp }& 0\end{array}\right]\\ =r\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ \stackrel{^}{X}& 0& 0\\ N& A& 0\end{array}\right]-r\left(\stackrel{^}{X}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{by}\text{\hspace{0.17em}}\left(11\right)\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}{X}^{\perp }={F}_{{X}^{\prime }}\right)\\ =r\left[\begin{array}{cc}{F}^{\perp }V{\stackrel{^}{X}}^{\perp }& {F}^{\perp }\stackrel{^}{X}\\ N{\stackrel{^}{X}}^{\perp }& A\end{array}\right]+r\left(F\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\text{by}\text{\hspace{0.17em}}\left(12\right)\right)\end{array}$

$\begin{array}{l}r\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp },W,V{W}^{\perp }\right]\\ =r\left[\stackrel{^}{X},V{\stackrel{^}{X}}^{\perp },\stackrel{^}{X},0,V{W}^{\perp }\right]=r\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ \stackrel{^}{X}& 0& 0\end{array}\right]-r\left(\stackrel{^}{X}\right)\\ =r\left[{F}^{\perp }V{\stackrel{^}{X}}^{\perp },{F}^{\perp }\stackrel{^}{X}\right]+r\left(D\right)\end{array}$

(a) $BLU{E}_{{M}_{1}}\left(A\alpha \right)=BLU{E}_{{M}_{2}}\left(A\alpha \right)$

(b) $r\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\\ 0& A& 0\end{array}\right]=r\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\end{array}\right]$

(c) $r\left[\begin{array}{cc}{F}^{\perp }V{\stackrel{^}{X}}^{\perp }& {F}^{\perp }\stackrel{^}{X}\\ 0& A\end{array}\right]=r\left[{F}^{\perp }V{\stackrel{^}{X}}^{\perp },{F}^{\perp }\stackrel{^}{X}\right]$

(d) $R\left({\left[\begin{array}{ccc}0& A& 0\end{array}\right]}^{\prime }\right)\subseteq R\left({\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\end{array}\right]}^{\prime }\right)$

(e) $R\left({\left[0,A\right]}^{\prime }\right)\subseteq R\left({\left[{F}^{\perp }V{\stackrel{^}{X}}^{\perp },{F}^{\perp }\stackrel{^}{X}\right]}^{\prime }\right)$

(f) 特别地，下面结论等价。

(I) 这里存在 $BLU{E}_{{M}_{1}}\left(XZ\alpha \right)$满足 $BLU{E}_{{M}_{1}}\left(XZ\alpha \right)=BLU{E}_{{M}_{2}}\left(XZ\alpha \right)$

(II) $r\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\\ 0& \stackrel{^}{X}& 0\end{array}\right]=r\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\end{array}\right]$

(III) $r\left[\begin{array}{cc}{F}^{\perp }V{\stackrel{^}{X}}^{\perp }& {F}^{\perp }\stackrel{^}{X}\\ 0& \stackrel{^}{X}\end{array}\right]=r\left[{F}^{\perp }V{\stackrel{^}{X}}^{\perp },{F}^{\perp }\stackrel{^}{X}\right]$

(IV) $R\left({\left[0,\stackrel{^}{X},0\right]}^{\prime }\right)\subseteq R\left({\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\end{array}\right]}^{\prime }\right)$

(V) $R\left({\left[0,\stackrel{^}{X}\right]}^{\prime }\right)\subseteq R\left({\left[{F}^{\perp }V{\stackrel{^}{X}}^{\perp },{F}^{\perp }\stackrel{^}{X}\right]}^{\prime }\right)$

(a) $BLU{P}_{{M}_{1}}\left(X\beta \right)=BLU{P}_{{M}_{2}}\left(X\beta \right)$

(b) $r\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\\ X\Sigma {\stackrel{˜}{X}}^{\prime }& \stackrel{^}{X}& 0\end{array}\right]=r\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\end{array}\right]$

(c) $r\left[\begin{array}{cc}{F}^{\perp }V{\stackrel{^}{X}}^{\perp }& {F}^{\perp }\stackrel{^}{X}\\ X\Sigma {\stackrel{˜}{X}}^{\prime }{\stackrel{^}{X}}^{\perp }& \stackrel{^}{X}\end{array}\right]=r\left[{F}^{\perp }V{\stackrel{^}{X}}^{\perp },{F}^{\perp }\stackrel{^}{X}\right]$

(d) $R\left({\left[X\Sigma {\stackrel{˜}{X}}^{\prime },\stackrel{^}{X},0\right]}^{\prime }\right)\subseteq R\left({\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\end{array}\right]}^{\prime }\right)$

(e) $R\left({\left[X\Sigma {\stackrel{˜}{X}}^{\prime }{\stackrel{^}{X}}^{\perp },\stackrel{^}{X}\right]}^{\prime }\right)\subseteq R\left({\left[{F}^{\perp }V{\stackrel{^}{X}}^{\perp },{F}^{\perp }\stackrel{^}{X}\right]}^{\prime }\right)$

(a) $BLU{P}_{{M}_{1}}\left(\epsilon \right)=BLU{P}_{{M}_{2}}\left(\epsilon \right)$

(b) $r\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\\ S\Sigma {\stackrel{^}{X}}^{\prime }& 0& 0\end{array}\right]=r\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\end{array}\right]$

(c) $r\left[\begin{array}{cc}{F}^{\perp }V{\stackrel{^}{X}}^{\perp }& {F}^{\perp }\stackrel{^}{X}\\ S\Sigma {\stackrel{^}{X}}^{\prime }{\stackrel{^}{X}}^{\perp }& 0\end{array}\right]=r\left[{F}^{\perp }V{\stackrel{^}{X}}^{\perp },{F}^{\perp }\stackrel{^}{X}\right]$

(d) $R\left({\left[S\Sigma {\stackrel{^}{X}}^{\prime },0,0\right]}^{\prime }\right)\subseteq R\left({\left[\begin{array}{ccc}V& \stackrel{^}{X}& F\\ {\stackrel{^}{X}}^{\prime }& 0& 0\end{array}\right]}^{\prime }\right)$

(e) $R\left({\left[S\Sigma {\stackrel{^}{X}}^{\prime }{\stackrel{^}{X}}^{\perp },0\right]}^{\prime }\right)\subseteq R\left({\left[{F}^{\perp }V{\stackrel{^}{X}}^{\perp },{F}^{\perp }\stackrel{^}{X}\right]}^{\prime }\right)$

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