Estimation of Population Means in Multivariate Stratified Random Sampling

In multivariate surveys where p (> 1) characteristics are defined on each unit of the population, the problem of allocation becomes complicated. In the present article, we propose a method to work out the compromise allocation in a multivariate stratified surveys. The problem is formulated as a Multiobjective Integer Nonlinear Programming Problem. Using the value function technique, the problem is converted into a single objective problem. A formula for continuous sample sizes is obtained using Lagrange Multipliers Technique (LMT) that can provide a near optimum solution in some cases. It may give an initial point for any integer nonlinear programing technique.     

On the Problem of Compromise Allocation in Multi-Response Stratified Sample Surveys

In stratified sample surveys, the problem of determining the optimum allocation is well known due to articles published in 1923 by Tschuprow and in 1934 by Neyman. The articles suggest the optimum sample sizes to be selected from each stratum for which sampling variance of the estimator is minimum for fixed total cost of the survey or the cost is minimum for a fixed precision of the estimator. If in a sample survey more than one characteristic is to be measured on each selected unit of the sample, that is, the survey is a multi-response survey, then the problem of determining the optimum sample sizes to various strata becomes more complex because of the non-availability of a single optimality criterion that suits all the characteristics. Many authors discussed compromise criterion that provides a compromise allocation, which is optimum for all characteristics, at least in some sense. Almost all of these authors worked out the compromise allocation by minimizing some function of the sampling variances of the estimators under a single cost constraint. A serious objection to this approach is that the variances are not unit free so that minimizing any function of variances may not be an appropriate objective to obtain a compromise allocation. This fact suggests the use of coefficient of variations instead of variances. In the present article, the problem of compromise allocation is formulated as a multi-objective non-linear programming problem. By linearizing the non-linear objective functions at their individual optima, the problem is approximated to an integer linear programming problem. Goal programming technique is then used to obtain a solution to the approximated problem.     

Impact of demographic variables on women’s economic empowerment: An ordered probit model

The present study aimed to empirically examine the demographic variables that determine women’s economic empowerment. A sample of 500 married women between 21 and 49 years old (M_age = 35.49, SD = 7.66) was conveniently selected from district Multan (Pakistan). Control over economic resources was used as a proxy for women’s economic empowerment. Ordered probit regression was run to assess the demographic determinants (i.e., age, education, paid job, income, and property) of economic empowerment of the least empowered, moderately empowered, and highly empowered women. Paid job, age, income, and property appeared as positive and significant predictors of women’s economic empowerment. Implications of the study were also discussed.     

Optimum multivariate stratified double sampling design: Chebyshev's Goal Programming approach

In stratified sampling when strata weights are unknown a double sampling technique may be used to estimate them. A large simple random sample from the unstratified population is drawn and units falling in each stratum are recorded. A stratified random sample is then selected and simple random subsamples are obtained out of the previously selected units of the strata. This procedure is called double sampling for stratification. If the problem of non-response is there, then subsamples are divided into classes of respondents and non-respondents. A second subsample is then obtained out of the non-respondents and an attempt is made to obtain the information by increasing efforts, persuasion and call backs. In this paper, the problem of obtaining a compromise allocation in multivariate stratified random sampling is discussed when strata weights are unknown and non-response is present. The problem turns out to be a multiobjective non-linear integer programming problem. An approximation of the problem to an integer linear programming problem by linearizing the non-linear objective functions at their individual optima is worked out. Chebyshev's goal programming technique is then used to solve the approximated problem. A numerical example is also presented to exhibit the practical application of the developed procedure.     

Optimum Multivariate Stratified Sampling Designs with Travel Cost: A Multiobjective Integer Nonlinear Programming Approach

The problem of optimum allocation in stratified sampling and its solution is well known in sampling literature for univariate populations (see Cochran, 1977 Cochran , W. G. ( 1977 ). Sampling Techniques. , 3rd ed. New York : Wiley . [Google Scholar]; Sukhatme et al., 1984 Sukhatme , P. V. , Sukhatme , B. V. , Sukhatme , S. , Ashok , C. ( 1984 ). Sampling Theory of Surveys With Applications. , 3rd ed. Ames , and New Delhi : Iowa State University Press and Indian Society of Agricultural Statistics . [Google Scholar]). In multivariate populations where more than one characteristics are to be studied on every selected unit of the population the problem of finding an optimum allocation becomes more complex due to conflicting behaviour of characteristics. Various authors such as Dalenius (1953 Dalenius , T. ( 1953 ). The multivariate sampling problem . Skandinavisk Actuarietidskrift 36 : 92 – 102 . [Google Scholar], 1957 Dalenius , T. ( 1957 ). Sampling in Sweden. 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Almost all the previous authors used some function of the sampling variances of the estimators of various characteristics to be measured as an objective that is to be minimized for a fixed cost given as a linear function of sample allocations. Because the variances are not unit free it is more logical to consider the minimization of some function of squared coefficient of variations as an objective. Previously this concept was used by Kozok (2006 Kozok , M. ( 2006 ). On sample allocation in multivariate surveys . Communication in Statistics—Simulation and Computation 35 : 901 – 910 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).

Furthermore, investigators have to approach the sampled units in order to get the observations. This involves some travel cost. Usually this cost is neglected while constructing a cost function. This travel cost may be significant in some surveys. For example if the strata consist of some geographically difficult-to-approach areas.

The authors problem of optimum allocation in multivariate stratified sampling is discussed with an objective to minimize simultaneously the coefficients of variation of the estimators of various characteristics under a cost constraint that includes the measurement as well as travel cost. The formulated problem of obtaining an optimum compromise allocation turns out to be a multiobjective all-integer nonlinear programming problem. Three different approaches are considered: the value function approach, ∈ –constraint method, and Distance–based method, to obtain compromise allocations. The cost function considered also includes the travel cost within stratum to reach the selected units. Additional restrictions are placed on the sample sizes to avoid oversampling and ensure the availability of the estimates of the strata variances. Numerical examples are also presented to illustrate the computational details of the proposed methods.