6. Combined estimators

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

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The strictly positive AML estimator of $A$  has typically a larger bias than ML or REML estimators for $A$  small relative to the $D_i’\text{s}\text{.}$  Thus, if we still wish to obtain a small area estimator that attaches a strictly positive weight to the direct estimator, to reduce the mentioned bias it will be better to use the AML estimator only when strictly necessary; that is, either when data does not provide enough evidence against $A=0$  or when the resulting REML estimator of $A$  is zero. This section introduces two small area estimators of $\theta$  that give a strictly positive weight to the direct estimator, which are obtained as a combination of the EBLUP based on the AML method and the EBLUP based on REML estimation.

In the first combined proposal, the AML method is used to estimate $A$  when the preliminary test does not reject the null hypothesis and in the second combined proposal, when the REML estimate is non positive. Specifically, the first combined estimator, called hereafter PT-AML, is defined by

$${\widehat{\theta }}_{\text{PTAML}}=\{\begin{array}{llll}{\widehat{\theta }}_{\text{AML}}\hfill & \text{if }T\le X_{m-p\mathrm{,}\alpha }^2\hfill & \text{or}\hfill & {\widehat{A}}_{\text{RE}}=0,\hfill \\ {\widehat{\theta }}_{\text{RE}}\hfill & \text{if }T>X_{m-p\mathrm{,}\alpha }^2\hfill & \text{and}\hfill & {\widehat{A}}_{\text{RE}}>0.\hfill \end{array}(6.1)$$

The second combined estimator, called REML-AML, is given by

$${\widehat{\theta }}_{\text{REAML}}=\{\begin{array}{ll}{\widehat{\theta }}_{\text{AML}}\hfill & \text{if }{\widehat{A}}_{\text{RE}}=0,\hfill \\ {\widehat{\theta }}_{\text{RE}}\hfill & \text{if }{\widehat{A}}_{\text{RE}}>0,\hfill \end{array}(6.2)$$

see Rubin-Bleuer and Yu (2013). For the estimation of MSE of ${\widehat{\theta }}_{\text{REAML}},$  these authors proposed

$$\text{mse}\left({\widehat{\theta }}_{\text{REAML}\mathrm{,}i}\right)=\{\begin{array}{ll}\text{mse}\left({\widehat{\theta }}_{\text{AML}\mathrm{,}i}\right)\hfill & \text{if }{\widehat{A}}_{\text{RE}}=0,\hfill \\ \text{mse}\left({\widehat{\theta }}_{\text{RE}\mathrm{,}i}\right)\hfill & \text{if }{\widehat{A}}_{\text{RE}}>0.\hfill \end{array}(6.3)$$

Using $\text{mse}\left({\widehat{\theta }}_{\text{AML}\mathrm{,}i}\right)$  when ${\widehat{A}}_{\text{RE}}=0$  leads to substantial overestimation if the true value of $A$  is small because ${\widehat{\theta }}_{\text{AML}\mathrm{,}i}$  will be closer to the regression-synthetic estimator. Hence, we propose the alternative MSE estimator

$${\text{mse}}_0\left({\widehat{\theta }}_{\text{REAML}\mathrm{,}i}\right)=\{\begin{array}{ll}{g}_{2i}\hfill & \text{if }{\widehat{A}}_{\text{RE}}=0,\hfill \\ \text{mse}\left({\widehat{\theta }}_{\text{RE}\mathrm{,}i}\right)\hfill & \text{if }{\widehat{A}}_{\text{RE}}>0.\hfill \end{array}(6.4)$$

Again, since for small $A,\text{mse}\left({\widehat{\theta }}_{\text{RE}\mathrm{,}i}\right)$  might still be overestimating the true MSE of ${\widehat{\theta }}_{\text{REAML}\mathrm{,}i},$  we consider also the following PT estimator

$${\text{mse}}_{\text{PT}}\left({\widehat{\theta }}_{\text{REAML}\mathrm{,}i}\right)=\{\begin{array}{llll}{g}_{2i}\hfill & \text{if }T\le X_{m-p\mathrm{,}\alpha }^2\hfill & \text{or}\hfill & {\widehat{A}}_{\text{RE}}=0,\hfill \\ \text{mse}\left({\widehat{\theta }}_{\text{RE}\mathrm{,}i}\right)\hfill & \text{if }T>X_{m-p\mathrm{,}\alpha }^2\hfill & \text{and}\hfill & {\widehat{A}}_{\text{RE}}>0.\hfill \end{array}(6.5)$$

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