Model-based small area estimation under informative sampling 5. Concluding remarksModel-based small area estimation under informative sampling 5. Concluding remarks

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In this paper, we studied model-based small area estimation for different levels of design informativeness under a nested error linear regression model for the population units. Estimators considered were the EBLUP, the pseudo-EBLUP (You and Rao 2002) and an estimator given by Pfeffermann and Sverchkov (2007). The EBLUP and the pseudo-EBLUP were computed under two scenarios: (i) Ignore informative sampling and assume that the population model holds for the sample; (ii) Take account of informative sampling by using a suitable function of the unit selection probability $p_{j|i}$ as an additional auxiliary variable in the sample model.

Results from a simulation study showed that design informativeness can have a big impact on the bias and MSE of the EBLUP that ignores informative sampling (scenario (i)). Results under scenario (ii) showed that the EBLUP, based on the augmented model, performs extremely well in terms of bias and MSE, provided that the augmenting variable is chosen properly. The bias-adjusted estimator of Pfeffermann and Sverchkov (2007) also performed well under informative sampling in terms of bias but its MSE is significantly larger than the corresponding MSE of the EBLUP and the pseudo-EBLUP based on the augmented model. Pseudo-EBLUP under scenario (i) performed significantly better than the corresponding EBLUP. It can be significantly improved by using the augmented model, similar to the case of EBLUP.

An advantage of the augmented model approach is that no new theory is required for estimation and MSE estimation. However, the area mean ${\overline{G}}_i$ of the augmenting variable $g_{ij}$ is required, unlike in the approach of Pfeffermann and Sverchkov (2007). For some choices of $g_{ij},{\overline{G}}_i$ is readily known; for example $g_{ij}=p_{j|i}$ gives ${\overline{G}}_i=1/N_i$ and $g_{ij}=n_i{w}_{j|i}$ gives ${\overline{G}}_i=n_i{\overline{W}}_i$ and ${\overline{W}}_i$ is often known for some surveys. We have also given a method of choosing the augmenting variable $g_{ij}.$

In this paper, we focused on the special case where all the areas are sampled. Extension of the augmented model approach to handle non-sampled areas requires the knowledge of the area means ${\overline{G}}_i,$ as well as the area selection probabilities, $p_i,$ for the non-sampled areas. This extension is currently under study.

Acknowledgements

We are thankful to the Associate Editor, the referees and M. Sverchkov for many constructive comments and suggestions.

References

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