On Constructing Optimum Strata and Determining Optimum Allocation

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 must be addressed by the sampler while using stratified sampling. There were several methods available to determine the OSB when the frequency distribution of the study (or its related) variable is known. 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 variable are known prior to conducting the survey. Moreover, as both the problems are not addressed simultaneously, the OSB and the sample allocation 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 its 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 variable, 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 software, LINGO. Two numerical examples, when the study variable or its auxiliary variable has respectively a uniform and a right-triangular distribution in the population, are presented to demonstrate the practical application of the proposed method and its computational details. The proposed technique can easily be applied to other frequency distributions