Small area quantile estimation via spline regression and empirical likelihood

This paper studies small area quantile estimation under a unit level non-parametric nested-error regression model. We assume the small area specific error distributions satisfy a semi-parametric density ratio model. We fit the non-parametric model via the penalized spline regression method of Opsomer, Claeskens, Ranalli, Kauermann and Breidt (2008). Empirical likelihood is then applied to estimate the parameters in the density ratio model based on the residuals. This leads to natural area-specific estimates of error distributions. A kernel method is then applied to obtain smoothed error distribution estimates. These estimates are then used for quantile estimation in two situations: one is where we only have knowledge of covariate power means at the population level, the other is where we have covariate values of all sample units in the population. Simulation experiments indicate that the proposed methods for small area quantiles estimation work well for quantiles around the median in the first situation, and for a broad range of the quantiles in the second situation. A bootstrap mean square error estimator of the proposed estimators is also investigated. An empirical example based on Canadian income data is included.