Determining Optimum Strata Boundaries and Sample Sizes for Skewed Population with Log-Normal Distribution

• Published in: Communications in statistics. Simulation and computation Vol. 44; no. 5; pp. 1364 - 1387
• Main Authors: Khan, M. G. M., Rao, Dinesh, Ansari, A. H., Ahsan, M. J.
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 28.05.2015; Taylor & Francis Ltd
• Subjects: Boundary layer; Dynamic programming technique; Frequency distribution; Log-normal distribution; Mathematical models; Multistage decision problem; Nonlinear programming problem; Optimization; Optimum sample size; Optimum stratification; Primary 62D05; Samples; Secondary 80M50; Statistical methods; Strata; Stratified sampling; Variance
• ISSN: 0361-0918; 1532-4141
• DOI: 10.1080/03610918.2013.819917

The method of choosing the best boundaries that make strata internally homogenous as far as possible is known as optimum stratification. To achieve this, the strata should be constructed in such a way that the strata variances for the characteristic under study be as small as possible. If the frequency distribution of the study variable x is known, the optimum strata boundaries (OSB) could be obtained by cutting the range of the distribution at suitable points. If the frequency distribution of x is unknown, it may be approximated from the past experience or some prior knowledge obtained at a recent study. Many skewed populations have log-normal frequency distribution or may be assumed to follow approximately log-normal frequency distribution. In this article, the problem of finding the OSB and the optimum sample sizes within the stratum for a skewed population with log-normal distribution is studied. The problem of determining the OSB is redefined as the problem of determining optimum strata widths (OSW) and is formulated as a Nonlinear Programming Problem (NLPP) that seeks minimization of the variance of the estimated population mean under Neyman allocation subject to the constraint that the sum of the widths of all the strata is equal to the range of the distribution. The formulated NLPP turns out to be a multistage decision problem that can be solved by dynamic programming technique. A numerical example is presented to illustrate the application and computational details of the proposed method. A comparison study is conducted to investigate the efficiency of the proposed method with other stratification methods, viz., Dalenius and Hodges' cum method, geometric method by Gunning and Horgan, and Lavallée-Hidiroglou method using Kozak's algorithm available in the literature. The study reveals that the proposed technique is efficient in minimizing the variance of the estimate of the population mean and is useful to obtain OSB for a skewed population with log-normal frequency distribution.