4. Preliminary test estimators

Isabel Molina, J.N.K. Rao and Gauri Sankar Datta

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The estimator of $A$  used in the EBLUP of ${\theta }_i$  introduces uncertainty, which might not be negligible for small $m.$  Indeed, the term $g_{3i}$  in the MSE estimator (3.2) arises due to the estimation of $A.$  However, when the value of $A$  is small enough relative to the sampling variances, this uncertainty could be avoided by using the regression-synthetic estimator ${x^{\prime }}_i\tilde{\beta }\left(0\right)$  instead of the EBLUP. Datta et al. (2011) proposed a small area estimator based on a preliminary testing procedure of $H_0:A=0$  against $H_1:A>0.$  When $H_0$  is not rejected, the regression-synthetic estimator is taken as the estimator of ${\theta }_i;$  otherwise, the usual EBLUP is used. They proposed the test statistic

$$T={\left(y-X{\widehat{\beta }}_{\text{PT}}\right)}^{\prime }{D}^{-1}\left(y-X{\widehat{\beta }}_{\text{PT}}\right)\mathrm{,}$$

where ${\widehat{\beta }}_{\text{PT}}={\left(X^{\prime }{D}^{-1}X\right)}^{-1}{X}^{\prime }{D}^{-1}y$  is the WLS estimator of $\beta$  obtained assuming that $H_0:A=0$  is true. The test statistic $T$  is distributed as $X_{m-p}^2$  with $m-p$  degrees of freedom under $H_0.$  Then, for a specified significance level $\alpha ,$  the PTE of $\theta$  defined by Datta et al. (2011) is given by

$${\widehat{\theta }}_{\text{PT}}={\left({\widehat{\theta }}_{\text{PT}\mathrm{,1}}\mathrm{,}\dots \mathrm{,}{\widehat{\theta }}_{\text{PT}\mathrm{,}m}\right)}^{\prime }=\{\begin{array}{ll}X{\widehat{\beta }}_{\text{PT}}\hfill & \text{if }T\le X_{m-p\mathrm{,}\alpha }^2;\hfill \\ {\widehat{\theta }}_{\text{RE}}\hfill & \text{if }T>X_{m-p\mathrm{,}\alpha }^2\mathrm{,}\hfill \end{array}$$

where $X_{m-p,\alpha }^2$  is the upper $\alpha -$point of $X_{m-p}^2.$  The PTE is especially designed to handle cases with a modest number of small areas, say $m=15.$

Here we propose to use the PT procedure for the estimation of MSE of the EBLUP, by considering only the MSE of the synthetic estimator $g_{2i}$  whenever the null hypothesis is not rejected and the full MSE estimate otherwise. But observe that the test statistic $T$  in the PT procedure does not depend on the estimator of $A.$  This means that, even when $H_0$  is rejected, it may happen that ${\widehat{A}}_{\text{RE}}=0.$  Thus, here we define the PT estimator of the MSE of the EBLUP ${\widehat{\theta }}_{\text{RE}\mathrm{,}i}$  as

$${\text{mse}}_{\text{PT}}\left({\widehat{\theta }}_{\text{RE}\mathrm{,}i}\right)=\{\begin{array}{llll}{g}_{2i}\hfill & \text{if }T\le X_{m-p\mathrm{,}\alpha }^2\hfill & \text{or}\hfill & {\widehat{A}}_{\text{RE}}=0,\hfill \\ g_{1i}\left({\widehat{A}}_{\text{RE}}\right)+g_{2i}\left({\widehat{A}}_{\text{RE}}\right)+2g_{3i}\left({\widehat{A}}_{\text{RE}}\right)\hfill & \text{if }T>X_{m-p\mathrm{,}\alpha }^2\hfill & \text{and}\hfill & {\widehat{A}}_{\text{RE}}>0.\hfill \end{array}(4.1)$$

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