Model-based non parametric regression estimation of the finite population total under two-stage cluster sampling

Abstract

A model-based nonparametric regression estimator for the finite population total under two-stage sampling is proposed. In stage one, a sample of clusters is obtained and in stage two, sub samples of elements within each selected cluster are obtained. It is assumed that auxiliary information is available for all the clusters in the population and that a non parametric model describes the relationship between the cluster total and the auxiliary variable. Under these conditions, the estimator for the finite population total due to Dorfman (1992) is adapted. The condition mean squared error of the proposed estimator given the auxiliary information is derived. Asymptotic properties of the estimator are studied by deriving its Asymptotic Mean Integrated Squared Error (AMISE). In particular the applicability to the choice of bandwidth which minimize AMISE is explored. It is shown that plug- in methods for estimating bandwidth based on the AMISE will be ineffective. In choosing the bandwidth, a method akin to the one used i n Dorfman(1992) for the direct element sampling situation is suggested. The results of an empirical study are used to compare the performance of the proposed estimator with that of the standard estimators of the total in current use. It is observed that the estimator performs better (in terms of efficiency) than the linear with the other non parametric estimators in most situations. It is also noted that the proposed estimator is easier to understand and implement than the local polynomial estimator which, in its present form, requires derivations of expressions of the inclusion probabilities which can be difficult to obtain especially when a complex design is involved.