Small area estimation of complex parameters under unit‐level models with skew‐normal errors

Abstract

Abstract The widely used Elbers–Lanjouw–Lanjouw (ELL) method of estimating complex parameters for areas with small sample sizes uses a fitted nested‐error model based on survey data to create simulated censuses of the variable of interest. The complex parameters obtained from each simulated censuses are then averaged to get the estimate. An empirical best (EB) method, under the nested‐error model with normal errors, is significantly more efficient, in terms of mean square error (MSE), than the ELL method when the normality assumption holds. However, it can perform poorly in terms of MSE when the model errors are not normally distributed. We relax normality by assuming skew‐normal errors, derive EB estimators, and study their MSE relative to EB based on normality and ELL. We propose bootstrap methods for MSE estimation. We also study an improvement to ELL by conditioning on the area random effects and without parametric assumptions on the errors.