Nonparametric regression estimation under complex sampling designs

The efficient use of auxiliary information to improve the precision of estimation of population quantities of interest is a central problem in survey sampling. We consider nonparametric regression estimation using much weaker assumptions on the superpopulation model in more general survey situations. Complex designs such as multistage and multiphase sampling are often employed in many large-scale surveys. Nonparametric model-assisted estimators, based on local polynomial regression, for two-stage and two-phase sampling designs are proposed. The local polynomial regression estimator is a nonparametric version of the generalized regression (GREG) estimator and shares most of the desirable properties of the generalized regression estimator. The estimator of the finite population total for two-stage element sampling with complete cluster auxiliary information is a linear combination of cluster total estimators, with sample-dependent weights that are calibrated to known control totals. The nonparametric estimator for two-phase sampling with a regression model for between-phase inference is also expressed as a weighted linear sum of the study variable of interest over a second-phase sample, in which the weights are not calibrated directly to known control totals, but are calibrated to the Horvitz-Thompson estimators of known control totals over a first-phase sample. Asymptotic design unbiasedness and design consistency of the estimators are established, and consistent variance estimators are proposed. Simulation experiments indicate that the local polynomial regression estimators are more efficient than parametric regression estimators under model misspecification, while being nearly as good when the parametric mean function is correctly specified.