The spatial empirical Bayes predictor of the small area mean for a lognormal variable of interest and spatially correlated random effects

Abstract

The standard small area estimator, the empirical best linear unbiased predictor (EBLUP), estimates small area parameters by way of linear mixed models. The EBLUP assumes normal and independent random small area effects as well as normal and independent random sampling errors. Under these assumptions, the variable of interest also follows a normal distribution. In practice, however, the above assumptions are often violated. The variable of interest is often non-normal and highly skewed, and the small areas are frequently spatially dependent. In this paper, we propose the spatial empirical Bayes predictor (SEBP) of the small area mean of a positively skewed variable of interest in the presence of spatial dependence among the random small area effects. We assume that the variable of interest follows a normal distribution after a log transformation and that its log transform is linked to some auxiliary variables by a nested error regression model. The SEBP is derived under the log-transformed nested error regression model. By way of simulation, we show that compared to its alternatives, i.e., the direct estimator which is solely based on the survey data for the small area under study, the EBLUP which does not take into account spatial dependence and skewness, the empirical Bayes predictor which takes into account skewness but not spatial dependence among the small areas, the SEBP has the smallest average relative bias and average relative root-mean-squared error for various combinations—though not all—of skewness and spatial correlation.