Abstract
The problem of constructing optimum stratum boundaries (OSB) and the problem of determining sample allocation to different strata are well known in the sampling literature. To increase the efficiency in the estimates of population parameters these problems are to be addressed by the sampler while using stratified sampling. There are several methods available to determine the OSB when the frequency distribution of the study (or a correlated) variable is available. Whereas, the problem of determining optimum allocation was addressed in the literature mostly as a separate problem assuming that the strata are already formed and the stratum variances are known. However, many of these attempts have been made with an unrealistic assumption that the frequency distribution and the stratum variances of the target population are known prior to conducting the survey. As both the problems are not addressed simultaneously, the OSB and the sample allocations so obtained may not be feasible or may be far from optimum.
In this paper, the problems of finding the OSB and the optimum allocation are discussed simultaneously when the population mean of the study variable y is of interest and a frequency distribution f ( y) or the frequency distribution f (x) of its auxiliary variable x is available. The problem is formulated as a Nonlinear Programming Problem (NLPP) that seeks minimization of the variance of the estimated population parameter of the target population, which is subjected to a fixed total sample size. The formulated NLPP is then solved by executing a program coded in a user’s friendly optimization software, LINGO. Two numerical examples, when the study variable or the auxiliary variable follows a uniform and a right-triangular distribution in the population, are presented to demonstrate the practical application of the proposed method or its computational details. The proposed technique can easily be applied to other frequency distributions.
Author: M. G. M. Khan and Sushita Sharma
Received on: March, 2015
Accepted on: May, 2015