Methods of Stratification for a Generalised Auxiliary Variable Optimum Allocation

Abstract

In stratified sampling, ever since Dalenius [1] undertook the problem of optimum stratification, the research in the area has been progressing in various perspectives and dimensions till date. Amidst the multifaceted developments in the trend of the research, consideration of the topic by taking into account various aspects such as different sample selection methods and allocations, study variable based stratification, auxiliary variable based stratification, superpopulation models, extension to two study variables for a single auxiliary variable, extension to two stratification variables for a single study variable etc., are a few noteworthy ones. However, with regard to considering optimum stratification of heteroscedastic populations, as live populations are generally heteroscedastic, it was Gupt and Ahamed [2,3] who considered the problem for a few allocations under a heteroscedastic regression superpopulation (HRS) model. As a sequel to the work of the authors, in this paper, the problem of optimum stratification for an objective variable y based on a concomitant variable x under the HRS model is considered for an allocation proposed by Gupt [4,5] and termed as Generalised Auxiliary Variable Optimum Allocation (GAVOA). Methods of stratification in the form of equations and approximate solutions to the equations which stratify populations at optimum strata boundaries (OSB) and approximately optimum strata boundaries (AOSB) respectively are obtained. Mathematical analysis is used in minimizing sampling variance of the estimator of population mean and deriving all the proposed methods of stratification. The proposed equations divide heteroscedastic populations, symmetrical or moderately skewed or highly skewed, at OSB, but, the equations are implicit in nature and not easy in solving. Therefore, a few methods of finding AOSB are deduced from the equations through analytically justified steps of approximation. The methods may provide practically feasible solutions in survey planning in stratifying heteroscedastic population of any level of heteroscedasticity and the work may contribute, to some extent, theoretically in the research area. The methods are empirically examined in a few generated heteroscedastic data of varied shapes with some assumed levels of heteroscedasticity and found to perform with high efficiency. The proposed methods of stratification are restricted to the particular allocation used.