5. Adjusted maximum likelihood

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

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The estimation methods for $A$  described in Section 2 might produce zero estimates. In this case, the EBLUPs will give zero weight to the direct estimators in all areas, regardless of the efficiency of the direct estimator in each area. On the other hand, survey sampling practitioners often prefer to give always a strictly positive weight to direct estimators because they are based on the area-specific unit level data for the variable of interest without the assumption of any regression model. For this situation, Li and Lahiri (2010) proposed the AML estimator that delivers a strictly positive estimator of $A.$  This estimator, denoted here ${\widehat{A}}_{\text{AML}},$  is obtained by maximizing the adjusted likelihood defined as

$$L_{\text{AML}}\left(A\right)=A\times L_P\left(A\right)\mathrm{.}$$

The EBLUP given in (2.6) with $\widehat{A}={\widehat{A}}_{\text{AML}}$  will be denoted hereafter as ${\widehat{\theta }}_{\text{AML}}={\left({\widehat{\theta }}_{\text{AML}\mathrm{,1}}\mathrm{,}\dots \mathrm{,}{\widehat{\theta }}_{\text{AML}\mathrm{,}m}\right)}^{\prime }.$  Note that ${\widehat{\theta }}_{\text{AML}}$  assigns strictly positive weights to direct estimators.

Li and Lahiri (2010) proposed a second order unbiased MSE estimator of ${\widehat{\theta }}_{\text{AML}\mathrm{,}i}$  given by

$$\begin{array}{lll}\text{mse}\left({\widehat{\theta }}_{\text{AML}\mathrm{,}i}\right)\hfill & =\hfill & g_{1i}\left({\widehat{A}}_{\text{AML}}\right)+g_{2i}\left({\widehat{A}}_{\text{AML}}\right)+2g_{3i}\left({\widehat{A}}_{\text{AML}}\right)\hfill \\ \hfill & -\hfill & B_i^2\left({\widehat{A}}_{\text{AML}}\right)b_{\text{AML}}\left({\widehat{A}}_{\text{AML}}\right)\mathrm{,}\hfill \end{array}(5.1)$$

where $b_{\text{AML}}\left(A\right)$  is the bias of ${\widehat{A}}_{\text{AML}}$  and it is given by

$$b_{\text{AML}}\left(A\right)=\frac{\text{trace}\left\{P\left(A\right)-{\Sigma }^{-1}\left(A\right)\right\}+2/A}{\text{trace}\left\{{\Sigma }^{-2}\left(A\right)\right\}}\mathrm{.}$$

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