Appendix A: Proof of Proposition 2.1
As the kernels of the univariate and multivariate normal distributions are
$$\begin{aligned} f(y|\mu ,\sigma ^2)= & {} \frac{1}{(2\pi \sigma ^2)^{1/2}}\exp \big \{-\frac{1}{2\sigma ^2}(y-\mu )^2)\big \}\propto \exp \big \{-\frac{1}{2\sigma ^2} y^2+\frac{\mu }{\sigma ^2}y\big \},\\ f(y|\mu ,\Sigma )= & {} \frac{1}{(2\pi )^{n/2}|\Sigma |^{1/2}}\exp \big \{-\frac{1}{2}(y-\mu )^{\prime }\Sigma ^{-1}(y-\mu )\big \}\\\propto & {} \exp \big \{-\frac{1}{2} y^{\prime }\Sigma ^{-1}y+\mu ^{\prime }\Sigma ^{-1}y\big \}, \end{aligned}$$
the following results holds.
Proof of ( 2.3 ) . The conditional distribution of \(\varvec{v}_d\) , given the estimators \(\varvec{x}_d\) and \(y_d\) , is
$$\begin{aligned} f(\varvec{v}_d|\varvec{x}_d,y_{d}) \varpropto&f(y_d|\varvec{x}_d,\varvec{v}_d) f(\varvec{v}_d) =\frac{1}{\sqrt{2\pi (\sigma _u^2+\sigma _d^2)}}\\&\exp \Big \{-\frac{1}{2(\sigma _u^2 +\sigma _d^2)}(y_d-\varvec{x}_d\varvec{\beta }-\varvec{v}_d^{\prime }\varvec{\beta })^2\Big \} \\&\cdot \frac{1}{(2\pi )^{p/2}|\varvec{\Sigma }_d|^{1/2}}\exp \Big \{-\frac{1}{2} \varvec{v}_d^{\prime }\varvec{\Sigma }_d^{-1}\varvec{v}_d\Big \} \\ \varpropto&\exp \Big \{-\frac{1}{2(\sigma _u^2+\sigma _d^2)} \big (\varvec{v}_d^{\prime }\varvec{\beta }\varvec{\beta }^{\prime }\varvec{v}_d-2\varvec{v}_d^{\prime } \varvec{\beta }(y_d-\varvec{x}_d\varvec{\beta })\Big \}\\&\exp \Big \{-\frac{1}{2}\varvec{v}_d^{\prime } \varvec{\Sigma }_d^{-1}\varvec{v}_d\Big \} \\ =&\exp \Big \{-\frac{1}{2}\varvec{v}_d^{\prime }\Big (\frac{\varvec{\beta }\varvec{\beta }^{\prime }}{\sigma _u^2+\sigma _d^2}+\varvec{\Sigma }_d^{-1}\Big )\varvec{v}_d +\frac{\varvec{v}_d^{\prime }\varvec{\beta }(y_d-\varvec{x}_d\varvec{\beta })}{\sigma _u^2+\sigma _d^2} \Big \}. \end{aligned}$$
Therefore, \(f(\varvec{v}_d|\varvec{x}_d,y_{d})\) is a multivariate normal distribution with parameters
$$\begin{aligned} \varvec{\Psi }_d=\text{ var }(\varvec{v}_d|\varvec{x}_d,y_{d})=\Big (\frac{\varvec{\beta }\varvec{\beta }^{\prime }}{\sigma _u^2+\sigma _d^2}+\varvec{\Sigma }_d^{-1}\Big )^{-1},\quad E[\varvec{v}_d|\varvec{x}_d,y_{d}]=\frac{y_d-\varvec{x}_d\varvec{\beta }}{\sigma _u^2+\sigma _d^2}\varvec{\Psi }_d\varvec{\beta }. \end{aligned}$$
For calculating \(\varvec{\Psi }_d\) , we apply the inversion formula
$$\begin{aligned} \big (A+uv^{\prime }\big )^{-1}=A^{-1}-\frac{A^{-1}uv^{\prime } A^{-1}}{1+v^{\prime } A^{-1}u}, \end{aligned}$$
with \(A=\varvec{\Sigma }_d^{-1}\) , \(u=\frac{1}{\sigma _u^2+\sigma _d^2}\,\varvec{\beta }\) , \(v^{\prime }=\varvec{\beta }^{\prime }\) . We have
$$\begin{aligned} \varvec{\Psi }_d=\varvec{\Sigma }_d-\frac{\frac{1}{\sigma _u^2+\sigma _d^2}\,\varvec{\Sigma }_d \varvec{\beta }\varvec{\beta }^{\prime } \varvec{\Sigma }_d}{1+\frac{1}{\sigma _u^2+\sigma _d^2} \,\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }} =\varvec{\Sigma }_d-\frac{\varvec{\Sigma }_d\varvec{\beta }\varvec{\beta }^{\prime } \varvec{\Sigma }_d}{\sigma _u^2+\sigma _d^2+\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }}. \end{aligned}$$
Therefore
$$\begin{aligned} \hat{\varvec{v}}_d^{bp}= & {} E[\varvec{v}_d|\varvec{x}_d,y_{d}]=\frac{y_d-\varvec{x}_d\varvec{\beta }}{\sigma _u^2+\sigma _d^2}\Big (\varvec{\Sigma }_d-\frac{\varvec{\Sigma }_d\varvec{\beta }\varvec{\beta }^{\prime } \varvec{\Sigma }_d}{\sigma _u^2+\sigma _d^2+\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }}\Big )\varvec{\beta }\\= & {} \frac{y_d-\varvec{x}_d\varvec{\beta }}{\sigma _u^2+\sigma _d^2}\Big (\varvec{\Sigma }_d\varvec{\beta }-\frac{\varvec{\Sigma }_d\varvec{\beta }(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta })}{\sigma _u^2 +\sigma _d^2+\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }}\Big )\\= & {} \frac{y_d-\varvec{x}_d\varvec{\beta }}{\sigma _u^2+\sigma _d^2} \Big (1-\frac{\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }}{\sigma _u^2+\sigma _d^2 +\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }}\Big )\varvec{\Sigma }_d\varvec{\beta }\\= & {} \frac{y_d-\varvec{x}_d\varvec{\beta }}{\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2}\,\varvec{\Sigma }_d\varvec{\beta }. \end{aligned}$$
\(\square \)
Proof of ( 2.4 ). The conditional distribution of \(u_d\) , given \(\varvec{x}_d\) and \(y_d\) , is
$$\begin{aligned} f(u_d|\varvec{x}_d,y_{d})\varpropto&f(u_d)f(y_d|\varvec{x}_d,u_d)= \frac{1}{(2\pi \sigma _u^2)^{1/2}}\exp \Big \{-\frac{1}{2\sigma _u^2}u_d^2\Big \}\\&\cdot \frac{1}{\sqrt{2\pi (\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _d^2)}} \exp \Big \{-\frac{1}{2(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _d^2)} (y_d-\varvec{x}_d\varvec{\beta }-u_d)^2\Big \} \\ \varpropto&\exp \Big \{-\frac{1}{2\sigma _u^2}u_d^2\Big \} \exp \Big \{-\frac{1}{2(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _d^2)} \big (u_d^2-2u_d(y_d-\varvec{x}_d\varvec{\beta })\big )\Big \} \\ =&\exp \Big \{-\frac{1}{2}\Big (\frac{1}{\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }+\sigma _d^2}+\frac{1}{\sigma _u^2}\Big )u_d^2 +\frac{y_d-\varvec{x}_d\varvec{\beta }}{\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _d^2}\,u_d\Big \} \\ =&\exp \left\{ -\frac{1}{2}\frac{1}{\frac{\sigma _u^2\big (\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }+\sigma _d^2\big )}{\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2 +\sigma _d^2}}\,u_d^2 + \frac{1}{\frac{\sigma _u^2\big (\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }+\sigma _d^2\big )}{\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2}} \frac{\sigma _u^2\big (y_d-\varvec{x}_d\varvec{\beta }\big )}{\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2} \,u_d\right\} . \end{aligned}$$
Therefore, \(f(u_d|\varvec{x}_d,y_d)\) is a univariate normal distribution with parameters
$$\begin{aligned} \text{ var }(u_d|\varvec{x}_d,y_{d})= & {} \frac{\sigma _u^2\big (\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }+\sigma _d^2\big )}{\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2},\\ \hat{u}_d^{bp}=E[u_d|\varvec{x}_d,y_{d}]= & {} \frac{\sigma _u^2}{\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2 +\sigma _d^2}\big (y_d-\varvec{x}_d\varvec{\beta }\big ). \end{aligned}$$
\(\square \)
Proof of (
2.5
).
$$\begin{aligned} \hat{\mu }_d^{bp}= & {} \varvec{x}_d\varvec{\beta }+E[\varvec{v}_d^{\prime }|\varvec{x}_d,y_d]\varvec{\beta }+E[u_d|\varvec{x}_d,y_d]\\= & {} \varvec{x}_d\varvec{\beta }+\frac{\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }}{\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2}\,\big (y_d-\varvec{x}_d\varvec{\beta }\big ) +\frac{\sigma _u^2}{\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2}\big (y_d-\varvec{x}_d\varvec{\beta }\big ) \\= & {} \varvec{x}_d\varvec{\beta }+\frac{\sigma _u^2+\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }}{\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2}\,\big (y_d-\varvec{x}_d\varvec{\beta }\big )\\= & {} \frac{\sigma _u^2+\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }}{\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2}\,y_d +\frac{\sigma _d^2}{\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2}\,\varvec{x}_d\varvec{\beta }. \end{aligned}$$
Appendix B: MSE calculations Under model (2.2 ), the mean squared error of \(\hat{\mu }^\mathrm{FHblup}_d\) is
$$\begin{aligned}&\text {MSE}(\hat{\mu }^\mathrm{FHblup}_d)\\&\quad =E\big [\big (\hat{\mu }^\mathrm{FHblup}_d-\mu _d\big )^2\big ] =E\big [\big (g_d y_d+(1-g_d)\varvec{x}_d\varvec{\beta }-\varvec{x}_d\varvec{\beta }-\varvec{v}_d^{\prime }\varvec{\beta }-u_d\big )^2\big ]\\&\quad =E\big [\big (g_d(y_d-\varvec{x}_d\varvec{\beta }-\varvec{v}_d^{\prime }\varvec{\beta }-u_d) -(1-g_d)(\varvec{v}_d^{\prime }\varvec{\beta }+u_d)\big )^2\big ] \\&\quad =E\big [\big (g_de_d+(1-g_d)(\varvec{v}_d^{\prime }\varvec{\beta }+u_d)\big )^2\big ] =g_d^2E[e_d^2]+(1-g_d)^2E[(\varvec{v}_d^{\prime }\varvec{\beta }+u_d)^2]\\&\quad =g_d^2\sigma _d^2+(1-g_d)^2(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2) = \frac{\sigma _u^4\sigma _d^2}{(\sigma _u^2+\sigma _d^2)^2} +\frac{(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2)\sigma _d^4}{(\sigma _u^2+\sigma _d^2)^2} \\&\quad = \frac{\sigma _u^2\sigma _d^2}{\sigma _u^2+\sigma _d^2} \left[ \frac{\sigma _u^2}{\sigma _u^2+\sigma _d^2} +\frac{\sigma _d^2}{\sigma _u^2+\sigma _d^2} \frac{\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2}{\sigma _u^2}\right] \ge g_d\sigma _d^2 =\text {MSE}_{FH}(\hat{\mu }^\mathrm{FHblup}_d). \end{aligned}$$
The conditional MSE of \(\hat{\mu }^{bp}_d\) with known \(\beta \) and unknown \(\sigma _u^2\) is
$$\begin{aligned}&\text {MSE}(\hat{\mu }^{bp}_d|\varvec{X})\\&\quad =E\big [\big (\hat{\mu }^{bp}_d-\mu _d\big )^2\big ] =E\big [\big (\gamma _d y_d+(1-\gamma _d)\varvec{x}_d\varvec{\beta }-\varvec{x}_d\varvec{\beta }-\varvec{v}_d^{\prime } \varvec{\beta }-u_d\big )^2\big ] \\&\quad =E\big [\big (\gamma _d(y_d-\varvec{x}_d\varvec{\beta }-\varvec{v}_d^{\prime }\varvec{\beta }-u_d)-(1-\gamma _d) (\varvec{v}_d^{\prime }\varvec{\beta }+u_d)\big )^2\big ] \\&\quad =E\big [\big (\gamma _de_d+(1-\gamma _d)(\varvec{v}_d^{\prime }\varvec{\beta }+u_d)\big )^2\big ] =\gamma _d^2E[e_d^2]+(1-\gamma _d)^2E[(\varvec{v}_d^{\prime }\varvec{\beta }+u_d)^2]\\&\quad =\gamma _d^2\sigma _d^2+(1-\gamma _d)^2(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2) =\frac{(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2)^2\sigma _d^2}{(\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2)^2} +\frac{(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2)\sigma _d^4}{(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2)^2}\\&\quad = \frac{(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2)\sigma _d^2(\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2)}{(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2 +\sigma _d^2)^2} = \frac{(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2)\sigma _d^2}{(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }+\sigma _u^2+\sigma _d^2)}=\gamma _d\sigma _d^2. \end{aligned}$$
The covariance matrices appearing in the derivation of the MSE of the BP \(\hat{\mu }^{bp}_d\) are
$$\begin{aligned} \varvec{R}_{11}\triangleq&E\left[ (\hat{\varvec{\beta }}-\varvec{\beta })(\hat{\varvec{\beta }}-\varvec{\beta })^{\prime }\right] =\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1}\varvec{V}\varvec{V}^{-1}\varvec{X}\varvec{Q}\\ =&\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1}\varvec{X}\varvec{Q}=\varvec{Q}\varvec{Q}^{-1}\varvec{Q}=\varvec{Q}, \\ \varvec{R}_{12}\triangleq&E\big [(\hat{\varvec{\beta }}-\varvec{\beta })(\hat{\varvec{u}}-\varvec{u})^{\prime }\big ] =E\big [\hat{\varvec{\beta }}\hat{\varvec{u}}^{\prime }\big ]-E\big [\hat{\varvec{\beta }}\varvec{u}^{\prime }\big ]\\ =&E\big [\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1}\varvec{y}\varvec{y}^{\prime }(\varvec{I}-\varvec{V}^{-1}\varvec{X}\varvec{Q}\varvec{X}^{\prime }) \varvec{V}^{-1}\varvec{V}_u\big ] -E\big [\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1}\varvec{y}\varvec{u}^{\prime }\big ]\\ =&\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1}\varvec{V}(\varvec{I}-\varvec{V}^{-1}\varvec{X}\varvec{Q}\varvec{X}^{\prime })\varvec{V}^{-1}\varvec{V}_u -\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1}\varvec{V}_u\\ =&-\varvec{Q}(\varvec{X}^{\prime }\varvec{V}^{-1}\varvec{X})\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1}\varvec{V}_u = -\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1}\varvec{V}_u, \\ \varvec{R}_{13}\triangleq&E\big [(\hat{\varvec{\beta }}-\varvec{\beta })(\hat{\varvec{w}}-\varvec{w})^{\prime }\big ] =-\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1}\underset{1\le d \le D}{\hbox {diag}}(\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta })\\ \varvec{R}_{22}\triangleq&E\big [(\hat{\varvec{u}}-\varvec{u})(\hat{\varvec{u}}-\varvec{u})^{\prime }\big ] = E\big [\hat{\varvec{u}}\hat{\varvec{u}}^{\prime }\big ]-E\big [\hat{\varvec{u}}\varvec{u}^{\prime }\big ] -E\big [\varvec{u}\hat{\varvec{u}}^{\prime }\big ]+E\big [\varvec{u}\varvec{u}^{\prime }\big ]\\ =&\varvec{V}_u\varvec{V}^{-1}(\varvec{I}-\varvec{X}\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1})\varvec{V}(\varvec{I}-\varvec{V}^{-1} \varvec{X}\varvec{Q}\varvec{X}^{\prime })\varvec{V}^{-1}\varvec{V}_u\\&- \varvec{V}_u\varvec{V}^{-1}(\varvec{I}-\varvec{X}\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1})\varvec{V}_u -\varvec{V}_u(\varvec{I}-\varvec{V}^{-1}\varvec{X}\varvec{Q}\varvec{X}^{\prime })\varvec{V}^{-1}\varvec{V}_u+\varvec{V}_u\\ =&\varvec{V}_u-\varvec{V}_u\varvec{V}^{-1}\varvec{V}_u+\varvec{V}_u\varvec{V}^{-1}\varvec{X}\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1}\varvec{V}_u,\\ \varvec{R}_{33}\triangleq&E\big [(\hat{\varvec{w}}-\varvec{w})(\hat{\varvec{w}}-\varvec{w})^{\prime }\big ] = E\big [\hat{\varvec{w}}\hat{\varvec{w}}^{\prime }\big ]-E\big [\hat{\varvec{w}}\varvec{w}^{\prime }\big ] -E\big [\varvec{w}\hat{\varvec{w}}^{\prime }\big ]+E\big [\varvec{w}\varvec{w}^{\prime }\big ]\\ =&\underset{1\le d \le D}{\hbox {diag}}(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }) -\underset{1\le d \le D}{\hbox {diag}}(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }) \varvec{V}^{-1}\underset{1\le d \le D}{\hbox {diag}}(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta })\\&+\underset{1\le d \le D}{\hbox {diag}}(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta })\varvec{V}^{-1} \varvec{X}\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1}\underset{1\le d \le D}{\hbox {diag}}(\varvec{\beta }^{\prime } \varvec{\Sigma }_d\varvec{\beta }),\\ \varvec{R}_{23}\triangleq&E\big [(\hat{\varvec{u}}-\varvec{u})(\hat{\varvec{w}}-\varvec{w})^{\prime }\big ] = E\big [\hat{\varvec{u}}\hat{\varvec{w}}^{\prime }\big ]-E\big [\hat{\varvec{u}}\varvec{w}^{\prime }\big ] -E\big [\varvec{u}\hat{\varvec{w}}^{\prime }\big ]\\ =&\varvec{V}_u\varvec{V}^{-1}(\varvec{I}-\varvec{X}\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1})\varvec{V}(\varvec{I}-\varvec{V}^{-1} \varvec{X}\varvec{Q}\varvec{X}^{\prime })\varvec{V}^{-1}\underset{1\le d \le D}{\hbox {diag}} (\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta })\\&-\varvec{V}_u\varvec{V}^{-1}(\varvec{I}-\varvec{X}\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1}) \underset{1\le d \le D}{\hbox {diag}}(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta })\\&-\varvec{V}_u(\varvec{I}-\varvec{V}^{-1}\varvec{X}\varvec{Q}\varvec{X}^{\prime })\varvec{V}^{-1}\underset{1\le d \le D}{\hbox {diag}}(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta })\\ =&-\varvec{V}_u\varvec{V}^{-1}\underset{1\le d \le D}{\hbox {diag}} (\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }) +\varvec{V}_u\varvec{V}^{-1}\varvec{X}\varvec{Q}\varvec{X}^{\prime }\varvec{V}^{-1} \underset{1\le d \le D}{\hbox {diag}}(\varvec{\beta }^{\prime }\varvec{\Sigma }_d\varvec{\beta }). \end{aligned}$$
Therefore, we have
$$\begin{aligned} \text {MSE}(\hat{\mu }_d^{bp}|\varvec{X})= & {} \varvec{a}_d^{\prime }\varvec{X}\varvec{R}_{11}\varvec{X}^{\prime }\varvec{a}_d +\varvec{a}_d^{\prime }\varvec{X}\varvec{R}_{12}\varvec{a}_d+\varvec{a}_d^{\prime }\varvec{X}\varvec{R}_{13}\varvec{a}_d +\varvec{a}_d^{\prime }\varvec{R}_{21}\varvec{X}^{\prime }\varvec{a}_d \\&+\varvec{a}_d^{\prime }\varvec{R}_{22}\varvec{a}_d+\varvec{a}_d^{\prime }\varvec{R}_{23}\varvec{a}_d +\varvec{a}_d^{\prime }\varvec{R}_{31}\varvec{X}^{\prime }\varvec{a}_d+\varvec{a}_d^{\prime }\varvec{R}_{32}\varvec{a}_d +\varvec{a}_d^{\prime }\varvec{R}_{33}\varvec{a}_d \\= & {} \varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime }-\varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime }\frac{\sigma _u^2}{b_d} -\varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime }\frac{a_d}{b_d} -\varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime } \frac{\sigma _u^2}{b_d}\\&+\Big (\sigma _u^2-\frac{\sigma _u^4}{b_d} +\frac{\sigma _u^4}{b_d^2}\varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime }\Big ) \\&+ \Big (-\frac{\sigma _u^2a_d}{b_d}+\frac{\sigma _u^2a_d}{b_d^2} \varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime }\Big ) -\varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime }\frac{a_d}{b_d}\\&+\Big (-\frac{\sigma _u^2a_d}{b_d}+\frac{\sigma _u^2a_d}{b_d^2} \varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime }\Big )\\+ & {} \Big (a_d-\frac{a_d^2}{b_d}+\frac{a_d^2}{b_d^2}\varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime }\Big ). \end{aligned}$$
Further simplification yields:
$$\begin{aligned} \text {MSE}(\hat{\mu }_d^{bp}|\varvec{X})= & {} g_{1d}(\varvec{\beta },\sigma _u^2)+g_{2d}(\varvec{\beta },\sigma _u^2),\\ g_{1d}(\varvec{\beta },\sigma _u^2)= & {} \sigma _u^2-\frac{\sigma _u^4}{b_d}-2 \frac{\sigma _u^2a_d}{b_d}+a_d-\frac{a_d^2}{b_d} = \sigma _u^2+a_d-\frac{\sigma _u^4+a_d^2+2\sigma _u^2a_d}{b_d}\\= & {} (\sigma _u^2+a_d)-\frac{(\sigma _u^2+a_d)^2}{(\sigma _u^2+a_d)+\sigma _d^2} =\frac{\sigma _d^2(\sigma _u^2+a_d)}{\sigma _u^2+a_d+\sigma _d^2}=\gamma _d\sigma _d^2,\\ g_{2d}(\varvec{\beta },\sigma _u^2)= & {} \varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime }\Big (1-2\frac{\sigma _u^2}{b_d} -2\frac{a_d}{b_d}+\frac{\sigma _u^4}{b_d^2}+\frac{a_d^2}{b_d^2} +2\frac{\sigma _u^2a_d}{b_d^2}\Big ) \\= & {} \varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime }\frac{1}{b_d^2}\big [b_d^2-2\sigma _u^2b_d -2a_db_d+\sigma _u^4+a_d^2+2\sigma _u^2a_d\big ] \\= & {} \varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime }\frac{1}{b_d^2}\big [(\sigma _u^4+a_d^2 +\sigma _d^4+2\sigma _u^2\sigma _d^2+2\sigma _u^2a_d+2a_d\sigma _d^2)\\&-2(\sigma _u^4+\sigma _u^2a_d+\sigma _u^2\sigma _d)\\&-2(a_d\sigma _u^2-a_d^2-a_d\sigma _d^2)+\sigma _u^4+a_d^2+2\sigma _u^2a_d\big ]\\= & {} \frac{\sigma _d^4}{(a_d+\sigma _u^2+\sigma _d^2)^2}\,\varvec{x}_d\varvec{Q}\varvec{x}_d^{\prime }. \end{aligned}$$
Appendix C See Tables 4 and 5 .
Table 4 Province poverty proportions for men (top) and women (bottom) based on the SLCS of 2008 Table 5 Province means (left) and std.errors (right) of covariates for men (top) and women (bottom) based on the SLFS of 2008