Optimal Predictors of General Small Area Parameters Under an Informative Sample Design Using Parametric Sample Distribution Models

Two challenges in small area estimation occur when (i) the sample selection mechanism depends on the outcome variable and (ii) the parameter of interest is a nonlinear function of the response variable in the assumed model. If, given the values of the model covariates, the sample selection mechanism depends on the model response variable, the design is said to be informative for the model. Pfeffermann and Sverchkov (2007) develop a small area estimation procedure for informative sampling, focusing on the prediction of small area means. Molina and Rao (2010) develop a small area estimation procedure for general parameters that are nonlinear functions of the model response variable. The method of Molina and Rao assumes noninformative sampling. We combine these two approaches to develop a procedure for the estimation of general parameters in small areas under informative sampling. We introduce a parametric bootstrap MSE estimator that is appropriate for an informative sample design. We evaluate the validity of the proposed procedures through extensive simulation studies and illustrate the procedures utilizing Mexico's income data.