Stratification Methods for an Auxiliary Variable Model-Based Allocation under a Superpopulation Model

Abstract

In this paper, the problem of optimum stratification of heteroscedastic populations in stratified sampling is considered for a known allocation under Simple Random Sampling With and Without Replacement (SRSWR & SRSWOR) design. The known allocation used in the problem is one of the model-based allocations proposed by Gupt [1,2] under a superpopulation model considered by Hanurav [3], Rao [4], and Gupt and Rao [5] which was modified by the author (Gupt [1,2]) to a more general form. The problem of finding optimum boundary points of stratification (OBPS) in stratifying populations considered here is based on an auxiliary variable which is highly correlated with the study variable. Equations giving the OBPS have been derived by minimizing the variance of estimator of the population mean. Since the equations giving OBPS are implicit and difficult for solving, some methods of finding approximately optimum boundary points of stratification (AOBPS) have also been obtained as the solutions of the equations giving OBPS. While deriving equations giving OBPS and methods of finding AOBPS, basic statistical definitions, tools of calculus, analytic functions and tools of algebra are used. While examining the efficiencies of the proposed methods of stratification, they are tested in a few generated populations and a live population. All the proposed methods of stratification are found to be efficient and suitable for practical applications. In this study, although the proposed methods are obtained under a heteroscedastic superpopulation model for level of heteroscedasticity one, the methods have shown robustness in empirical investigation in varied levels of heteroscedastic populations. The stratification methods proposed here are new as they are derived for an allocation, under the superpopulation model, which has never been used earlier by any researcher in the field of construction of strata in stratified sampling. The proposed methods may be a fascinating piece of work for researchers amidst the vigorously progressing theoretical research in the area of stratified sampling. Besides, by virtue of exhibiting high efficiencies in the performance of the methods, the work may provide a practically feasible solution in the planning of socio-economic survey.