8. Conclusions

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

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The following major conclusions may be drawn from the results of our simulation study on the estimation of small area means, based on the Fay-Herriot area-level model when the number of areas is modest in size $\left(\text{say }m=15\right):$  1) Under the Fay-Herriot model with a value of random effects variance, $A,$  clearly away from zero, the PTE does not seem to noticeably improve efficiency relative to the usual EBLUP unless the significance level is taken small $(\alpha \le 0.1$  in our simulation study). 2) Our simulation results indicate that using the PT procedure with a moderate $\alpha ,$  in particular $\alpha =0.2,$  to estimate the MSE of the usual EBLUP leads to a reduction in bias as compared with the usual MSE estimator. Hence, we recommend the use of ${\text{mse}}_{\text{PT}}\left({\widehat{\theta }}_{\text{RE}\mathrm{,}i}\right),$  given by (4.1), to estimate the MSE of the EBLUP. 3) Among the estimators that attach a strictly positive weight to the direct estimator for all areas, we recommend the combined estimator REML-AML given by (6.2), because it achieves slightly higher efficiency than the EBLUP based on AML and the PT-AML given by (6.1). 4) For estimating the MSE of the recommended REML-AML estimator, the estimator ${\text{mse}}_{\text{PT}}\left({\widehat{\theta }}_{\text{REAML}\mathrm{,}i}\right)$  given by (6.5) performs better than the alternative ones. 5) Our results on prediction intervals, based on normal theory, indicate that the good performance of the proposed MSE estimators may not translate to coverage properties of these intervals. Construction of prediction intervals that lead to accurate coverages, using the proposed MSE estimates, appears to be a difficult task.

Smooth alternatives to the preliminary test estimates in the case of location parameters have been proposed in the literature using weighted means of the estimates obtained under the null and alternative hypotheses, with weights depending on the test statistic, see e.g., Saleh (2006). Mean squared error estimates of this kind have not been studied and we leave this subject for further research.

Acknowledgements

We would like to thank the editor for very constructive suggestions. Gauri S. Datta’s research was partially supported through the grant H98230-11-1-0208 from the National Security Agency, Isabel Molina’s research by grants ref. MTM2009-09473, MTM2012-37077-C02-01 and SEJ2007-64500 from the Spanish Ministerio de Educación y Ciencia and J.N.K. Rao’s research by the Natural Sciences and Engineering Research Council of Canada.

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