Small area estimation techniques typically rely on regression models that use both covariates and random effects to explain between domain variation. Chambers and Tzavidis (2006) describe a novel approach to small area estimation that is based on modelling quantile-like parameters of the conditional distribution of the target variable given the covariates. This is an outlier robust approach that avoids conventional Gaussian assumptions and the problems associated with specification of random effects, allowing inter-domain differences to be characterized by the variation of area-specific M-quantile coefficients. These authors observed, however, that M-quantile estimates of small area means are biased with the magnitude of the bias being related to the presence of outliers in the data. In this paper we propose a bias adjustment to the M-quantile small area estimator of the mean that is based on representing this estimator as a functional of the small area distribution function. The method is then generalized for estimating other quantiles of the distribution function in a small area. The effect of this bias adjustment on small area estimation with random effects models in the presence of model misspecification is also examined.
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