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A Fay–Herriot model when auxiliary variables are measured with error

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Abstract

The Fay–Herriot model is an area-level linear mixed model that is widely used for estimating the domain means of a given target variable. Under this model, the dependent variable is a direct estimator calculated by using the survey data and the auxiliary variables are true domain means obtained from external data sources. Administrative registers do not always give good auxiliary variables so that statisticians sometimes take them from alternative surveys and therefore they are measured with error. We introduce a variant of the Fay–Herriot model that takes into account the measurement error of the auxiliary variables and give two fitting algorithms that calculate maximum and residual maximum likelihood estimates of the model parameters. Based on the new model, empirical best predictors of domain means are introduced and an approximation of its mean squared error is derived. We finally give an application to estimate poverty proportions in the Spanish Living Condition Survey, with auxiliary information from the Spanish Labour Force Survey.

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Author information

Correspondence to Jan Pablo Burgard.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported by the Spanish Grant MTM2015-64842-P and by the Grant “Algorithmic Optimization (ALOP) - graduate school 2126” funded by the German Research Foundation.

Appendices

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

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Burgard, J.P., Esteban, M.D., Morales, D. et al. A Fay–Herriot model when auxiliary variables are measured with error. TEST 29, 166–195 (2020). https://doi.org/10.1007/s11749-019-00649-3

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Keywords

  • Fay–Herriot model
  • Small area estimation
  • Measurement error
  • Monte Carlo simulation
  • Poverty proportion

Mathematics Subject Classification

  • 62F
  • 62P25
  • 62D05