Small area estimation using Fay-Herriot area level model with sampling variance smoothing and modeling
Section 2. Fay-Herriot model using EBLUP approach

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Under the Fay-Herriot model (1.3), assuming ${\sigma }_i^2$ and ${\sigma }_v^2$ known in the model, we obtain the best linear unbiased prediction (BLUP) estimator of ${\theta }_i$ as ${\tilde{\theta }}_i={\gamma }_i{y}_i+(1-{\gamma }_i){x_i^{\prime }}_{}\tilde{\beta },$ where ${\gamma }_i={\sigma }_v^2/({\sigma }_v^2+{\sigma }_i^2)$ and $\tilde{\beta }={\left({\displaystyle {\sum }_{i=1}^m{({\sigma }_i^2+{\sigma }_v^2)}^{-1}{x}_i{x_i^{\prime }}_{}}\right)}^{-1}\left({\displaystyle {\sum }_{i=1}^m{({\sigma }_i^2+{\sigma }_v^2)}^{-1}{x}_i{y}_i}\right).$ To estimate the variance component ${\sigma }_v^2,$ we have to first assume ${\sigma }_i^2$ known. There are several methods available to estimate ${\sigma }_v^2,$ and we use REML method to estimate ${\sigma }_v^2.$  Then the EBLUP of the small area parameter ${\theta }_i$ is obtained as

$${\widehat{\theta }}_i={\widehat{\gamma }}_i{y}_i+\left(1-{\widehat{\gamma }}_i\right){x_i^{\prime }}_{}\widehat{\beta },(2.1)$$

where ${\widehat{\gamma }}_i={\widehat{\sigma }}_v^2/\left({\widehat{\sigma }}_v^2+{\sigma }_i^2\right)$ and ${\widehat{\sigma }}_v^2$ is the REML estimator. The estimator for the mean squared error (MSE) of ${\widehat{\theta }}_i$ is given by mse $({\widehat{\theta }}_i)=g_{1i}+g_{2i}+2g_{3i},$ where $g_{1i}={\widehat{\gamma }}_i{\sigma }_i^2$ is the leading term, $g_{2i}$ accounts for the variability due to estimation of the regression parameter $\beta ,$ and $g_{3i}$ is due to the estimation of the model variance ${\sigma }_v^2;$  see Rao and Molina (2015) for details.

We may use the smoothed or direct estimate of ${\sigma }_i^2$ in (2.1). For sampling variance smoothing, we use a log-linear regression model on the direct sampling variance $s_i^2$ as suggested in You and Hidiroglou (2012), and the smoothing model is defined as:

$$\log (s_i^2)={\eta }_0+{\eta }_1\log (n_i)+{\epsilon }_i,i=1,\dots ,m,(2.2)$$

where the model error term is ${\epsilon }_i~N(0,{\psi }^2),$ and ${\psi }^2$ is unknown. Let ${\widehat{\eta }}_0$ and ${\widehat{\eta }}_1$ denote the ordinary least square estimates of the regression coefficients ${\eta }_0$ and ${\eta }_1,$ and ${\widehat{\psi }}^2$ be the estimated residual variance of the log-linear regression model (2.2). A smoothed estimator of the sampling variance ${\sigma }_i^2$ can be obtained as

$${\tilde{\sigma }}_i^2=\exp \left({\widehat{\eta }}_0+{\widehat{\eta }}_1\log (n_i)\right)exp\left({\widehat{\psi }}^2/2\right).$$

The smoothed sampling variances ${\tilde{\sigma }}_i^2$  can then be used in the EBLUP estimator (2.1) and its MSE computation. This procedure is a common practice, see Rao and Molina (2015).

If direct sampling variance estimate $s_i^2$ is used in the place of the true sampling variance ${\sigma }_i^2$ in (2.1), then an extra term accounting for the uncertainty of using $s_i^2$ is needed in the MSE estimator. This term, denoted as $g_{4i},$ is given as $g_{4i}=4{(n_i-1)}^{-1}{\widehat{\sigma }}_v^4{s}_i^4{({\widehat{\sigma }}_v^2+s_i^2)}^{-3};$ see Rivest and Vandal (2002) and Rao and Molina (2015), page 150. However, using $s_i^2$  directly in the EBLUP could lead to an over estimation of the model variance ${\sigma }_v^2$  (You, 2010; Rubin-Bleuer and You, 2016), as well as less accurate estimates. We will compare the EBLUP estimates with the HB estimates based on the smoothed and direct sampling variances in Section 4.

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