Constrained Small Area Estimators Based on M-quantile Methods

Small area estimators associated with M-quantile regression methods have been recently proposed by Chambers and Tzavidis (2006). These estimators do not rely on normality or other distributional assumptions, do not require explicit modelling of the random components of the model and are robust with respect to outliers and influential observations. In this article we consider two remaining problems which are relevant to practical applications. The first is benchmarking, that is the consistency of a collection of small area estimates with a reliable estimate obtained according to ordinary design-based methods for the union of the areas. The second is the correction of the under/over-shrinkage of small area estimators. In fact, it is often the case that, if we consider a collection of small area estimates, they misrepresent the variability of the underlying “ensemble” of population parameters. We propose benchmarked M-quantile estimators to solve the first problem, while for the second we propose an algorithm that is quite similar to the one used to obtain Constrained Empirical Bayes estimators, but that, consistently with the principles of M-estimation, does not make use of distributional assumptions and tries to achieve robustness with respect to the presence of outliers. The article is essentially about point estimation; we also introduce estimators of the mean squared error, but we do not deal with interval estimation.