1. Introduction
Isabel Molina, J.N.K. Rao and Gauri Sankar Datta
A basic area-level model, called the Fay-Herriot (FH) model, is often used to obtain efficient estimators of area means when the sample sizes within areas are small. This model involves unobservable area random effects, and the empirical best linear unbiased predictor (EBLUP) of a small area mean is obtained by estimating the associated random effect. The EBLUP is a weighted combination of a direct area-specific estimator and a regression-synthetic estimator that uses all the data. An estimator of the mean squared error (MSE) of the EBLUP was obtained first by Prasad and Rao (1990) using a moment estimator of the random effects variance and later by Datta and Lahiri (2000) for the restricted maximum likelihood (REML) estimator of the variance. Rao (2003, Chapter 7) gives a detailed account of EBLUPs and their MSE estimators for the FH model.
Sometimes the observed data do not support the inclusion of the area effects in the model. Excluding the area effects leads to the regression-synthetic estimator. Using this idea, Datta, Hall and Mandal (2011) proposed to do a preliminary test for the presence of the area random effects at a specified significance level, and then to define the small area estimator depending on the result of the test. If the null hypothesis of no area random effects is not rejected, the model without the area effects is considered to estimate the small area means, i.e., the regression-synthetic estimator is used. If the null hypothesis is rejected, the usual EBLUP under the FH model model with area effects is used. Datta et al. (2011) remarked that the above preliminary test estimator (PTE) could lead to significant efficiency gains over the EBLUP, particularly when the number of small areas is only modest in size. For preliminary testing, they considered a normality-based test as well as a bootstrap test that avoids the normality assumption.
When the estimated area effects variance is zero, the EBLUP becomes automatically the regression-synthetic estimator. However, the estimated MSE obtained by Prasad and Rao (1990) or Datta and Lahiri (2000) does not reduce to the estimated MSE of the regression-synthetic estimator. Thus, the usual MSE estimators are biased for small random effects variance. For this reason, we propose MSE estimators of the EBLUP based on the preliminary testing procedure. If the random effects variance is not significant according to the test, we consider the MSE estimator of the synthetic estimator. Otherwise, we consider the usual MSE estimators of the EBLUP.
The EBLUP attaches zero weight to the direct estimates for all areas when the estimated area effects variance is zero. On the other hand, survey practitioners often prefer to attach a strictly positive weight to the direct estimates, because the latter make use of the available area-specific unit level data and also incorporate the sampling design. Li and Lahiri (2010) introduced an adjusted maximum likelihood (AML) estimator of the variance of random effects that is always positive and therefore leads to EBLUPs giving strictly positive weights to direct estimators. As we shall see, a price is paid in terms of bias when using the EBLUP based on the AML estimator. We propose here alternative small area estimators that always give a positive weight to the direct estimators but with a smaller bias.
This paper studies empirically the properties of PTEs of small area means, in comparison with the usual EBLUPs and other proposed estimators. In particular, we study the choice of the significance level for the area estimates and for the MSE estimates based on the preliminary test (PT). EBLUPs based on the AML estimator of the random effects variance of Li and Lahiri (2010), which give non-zero weights to the direct estimators in all areas, are also studied and compared to PT versions of AML (PT-AML). Different MSE estimators of these PT-AML estimators are also studied with respect to relative bias. Based on simulation results, the EBLUPs and the associated MSE estimators that performed well are recommended. Finally, coverage and length of normality-based prediction intervals, obtained using the EBLUPs and the associated MSE estimators, are examined.
The paper is organized as follows. Section 2 describes the FH model model and the EBLUPs of small area means. Section 3 comments on MSE estimation. PTEs of small area means and MSE estimators based on the PT are introduced in Section 4. Section 5 describes small area estimators and associated MSE estimators under AML estimation of the area effects variance. Alternative estimators that also attach positive weights to direct estimators together with proposed MSE estimators are introduced in Section 6. Section 7 reports the results of the simulation study. Finally, Section 8 gives some concluding remarks.
- Date modified: