Comparison of some positive variance estimators for the Fay-Herriot small area model 2. EBLUP and MSE of the EBLUP under the Fay-Herriot modelComparison of some positive variance estimators for the Fay-Herriot small area model 2. EBLUP and MSE of the EBLUP under the Fay-Herriot model

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Let $y_i,i=1,\dots ,m,$ be the direct survey estimators of the small area means ${\theta }_i,i=1,\dots ,m.$ The Fay-Herriot model consists of the following sampling and linking models:

Sampling model: $y_i={\theta }_i+e_i,e_i|{\theta }_i\stackrel{\text{i}\text{.d}\text{.}}{\sim }\left(0,{\psi }_i\right),i=1,\dots ,m,(2.1)$

Linking model: ${\theta }_i=z_i^{\prime }\beta +v_i,v_i\stackrel{\text{i}\text{.i}\text{.d}\text{.}}{\sim }\left(0,{\sigma }_v^2\right),{\sigma }_v^2> 0,i=1,\dots ,m,(2.2)$

where $e_i$ are the sampling errors, independently distributed with mean zero and “known” sampling variances ${\psi }_i,$ $z_i\left(p\times 1\right)$ are known vectors of covariate values; $\beta$ is a $p\times 1$ vector of unknown fixed regression coefficients; and $v_i$ are independent and identically distributed random effects with mean zero and model variance ${\sigma }_v^2.$ Combining (2.1) and (2.2) we obtain:

$$y_i=z_i^{\prime }\beta +v_i+e_i,i=1,\dots ,m,(2.3)$$

with both model and sampling errors. The $y_i,i=1,\dots ,m,$ can be viewed as outcomes in the combined design-model space (see Rubin-Bleuer and Schiopu-Kratina 2005).

Under model (2.3), the EBLUP of the small area mean ${\theta }_i$ is given by:

$${\widehat{\theta }}_i\left({\widehat{\sigma }}_v^2\right)=z_i^{\prime }\widehat{\beta }\left({\widehat{\sigma }}_v^2\right)+{\widehat{\gamma }}_i\left[y_i-z_i^{\prime }\widehat{\beta }\left({\widehat{\sigma }}_v^2\right)\right]={\widehat{\gamma }}_i{y}_i+\left(1-{\widehat{\gamma }}_i\right)z_i^{\prime }\widehat{\beta }\left({\widehat{\sigma }}_v^2\right),i=1,\dots ,m,(2.4)$$

where ${\widehat{\sigma }}_v^2$ is a consistent estimator of ${\sigma }_v^2,$

$${\widehat{\gamma }}_i={\widehat{\sigma }}_v^2/\left({\widehat{\sigma }}_v^2+{\psi }_i\right),\text{ and }\widehat{\beta }\left({\widehat{\sigma }}_v^2\right)={\left[{\displaystyle \sum _{i=1}^m{z}_i{z}_i^{\prime }}/\left({\widehat{\sigma }}_v^2+{\psi }_i\right)\right]}^{-1}\left[{\displaystyle \sum _{i=1}^m{z}_i{y}_i}/\left({\widehat{\sigma }}_v^2+{\psi }_i\right)\right].(2.5)$$

To calculate the Mean Squared Error (MSE) of the EBLUP, we set the following regularity conditions:

1. The ${\psi }_i$ are bounded from above and away from zero,
2. The $z_i,1\le i\le m$ are bounded, and
3. $\lim \inf {\lambda }_{\min }\left(1/m{\displaystyle {\sum }_i{z}_i}\cdot z_i^{\prime }\right)>0$ where ${\lambda }_{\min }\left(A\right)=$ minimum eigenvalue of matrix $A.$

Under normality of the sampling errors $e_i$ associated with model (2.3) and the above regularity conditions, a second order approximation to the MSE is given by:

$$\text{MSE}\left\{{\widehat{\theta }}_i\left({\widehat{\sigma }}_v^2\right)\right\}=g_{1i}\left({\sigma }_v^2\right)+g_{2i}\left({\sigma }_v^2\right)+g_{3i}\left({\sigma }_v^2\right)+o\left(\frac{1}{m}\right),(2.6)$$

with $g_{1i}\left({\sigma }_v^2\right)={\gamma }_i{\psi }_i,$ $g_{2i}\left({\sigma }_v^2\right)={\left(1-{\gamma }_i\right)}^2{z}_i^{\prime }{\left[{\displaystyle {\sum }_{i=1}^m{z}_i{z}_i^{\prime }}/\left({\sigma }_v^2+{\psi }_i\right)\right]}^{-1}{z}_i$ and

$$g_{3i}\left({\sigma }_v^2\right)={\left({\psi }_i\right)}^2\overline{V}\left({\widehat{\sigma }}_v^2\right)/{\left({\sigma }_v^2+{\psi }_i\right)}^3,(2.7)$$

where $\overline{V}\left({\widehat{\sigma }}_v^2\right)$ is the asymptotic variance of ${\widehat{\sigma }}_v^2$ (Das, Jiang and Rao 2004).

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