Model Calibration at Different Stages in Cluster Sampling; Use of Penalized Splines in Semiparametric Estimation

Estimation of finite population total using internal calibration and model assistance on semiparametric models based on kernel methods have been considered by several authors. In this book chapter we extend this so as to consider model calibration based on penalized splines in two stage sampling where the auxiliary information is available both at the element level and at the cluster level. Specifically, we derive estimators of population total that incorporates model calibration in estimating the cluster totals and in estimation population total. We show that the proposed estimators are robust in the face of misspecified models, are asymptotic design unbiased, have reduced model bias, are design consistent and asymptotic normal. We have shown that estimators based on penalized splines perform better than corresponding kernel based estimators and model calibrated estimators perform better than internally calibrated estimators do.