M.J. Ahsan

Compromise allocation in multivariate stratified sampling : An integer solution

The problem of determining the sample sizes in various strata when several characteristics are under study is formulated as a nonlinear multistage decision problem. Dynamic programming is used to obtain an integer solution to the problem.

Theory & Methods: An Optimal Multivariate Stratified Sampling Design Using Dynamic Programming

Numerous optimization problems arise in survey designs. The problem of obtaining an optimal (or near optimal) sampling design can be formulated and solved as a mathematical programming problem. In multivariate stratified sample surveys usually it is not possible to use the individual optimum allocations for sample sizes to various strata for one reason or another. In such situations some criterion is needed to work out an allocation which is optimum for all characteristics in some sense. Such an allocation may be called an optimum compromise allocation. This paper examines the problem of determining an optimum compromise allocation in multivariate stratified random sampling, when the population means of several characteristics are to be estimated. Formulating the problem of allocation as an all integer nonlinear programming problem, the paper develops a solution procedure using a dynamic programming technique. The compromise allocation discussed is optimal in the sense that it minimizes a weighted sum of the sampling variances of the estimates of the population means of various characteristics under study. A numerical example illustrates the solution procedure and shows how it compares with Cochrans average allocation and proportional allocation.

A note on optimum allocation in multivariate stratified sampling

In stratified random sampling when several characteristics are to be estimated simultaneously, an allocation that is optimum for one characteristic may be far away from optimum for others. To resolve this conflict the authors formulate the problem of determining optimum compromise allocation as a nonlinear programming problem (NLPP). The allocation obtained is optimum in the sense that it minimizes the sum of weighted variances of the estimated population means of the characteristics subject to a fixed sampling cost. The formulated NLPP is treated as multistage decision problem and solved using dynamic programming technique. A numerical example is presented to illustrate the computational details.

Optimum stratification: a mathematical programming approach

The probelm of determining the optimum strata boundaries, when the main study variable is used as stratification variable and a stratified sample, using Neyman allocation (for a fixed total sample size) is to be selected to estimate the population mean (or total), is formulated as a mathematical programming problem (MPP). It has been shown that with some modification this MPP may be converted into a multistage decision problem that could be solved using dynamic programming technique. Two numerial examples are also presnted to illustrate the computational details.

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

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.

Estimation of more than one parameters in stratified sampling with fixed budget

In a multivariate stratified sampling more than one characteristic are defined on every unit of the population. An optimum allocation which is optimum for one characteristic will generally be far from optimum for others. A compromise criterion is needed to work out a usable allocation which is optimum, in some sense, for all the characteristics. When auxiliary information is also available the precision of the estimates of the parameters can be increased by using it. Furthermore, if the travel cost within the strata to approach the units selected in the sample is significant the cost function remains no more linear. In this paper an attempt has been made to obtain a compromise allocation based on minimization of individual coefficients of variation of the estimates of various characteristics, using auxiliary information and a nonlinear cost function with fixed budget. A new compromise criterion is suggested. The problem is formulated as a multiobjective all integer nonlinear programming problem. A solution procedure is also developed using goal programming technique.

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; Sukhatme et al., 1984). 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, 1957), Ghosh (1958), Yates (1960), Aoyama (1963), Gren (1964, 1966), Folks and Antle (1965), Hartley (1965), Kokan and Khan (1967), Chatterjee (1972), Ahsan and Khan (1977, 1982), Chromy (1987), Wywial (1988), Bethel (1989), Kreienbrock (1993), Jahan et al. (1994), Khan et al. (1997), Khan et al. (2003), Ahsan et al. (2005), Díaz-García and Ulloa (2006, 2008), Ahsan et al. (2009) etc. used different compromise criteria to work out a compromise allocation that is optimum for all characteristics in some sense. 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). 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.

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.

Multi-objective optimization for optimum allocation in multivariate stratified sampling with quadratic cost

In stratified sampling, usually the cost function is taken as a linear function of sample sizes n h . In many practical situations, the linear cost function does not approximate the actual cost incurred adequately. For example, when the cost of travelling between the units selected in the sample within a stratum is significant, instead of a linear cost function, a cost function that is quadratic in √n h will be a more close approximation to the actual cost. In this paper, the problem is formulated as multi-objective nonlinear integer programming problem with quadratic cost under three different situations, i.e. complete, partial or null information about the population. A numerical example is also presented to illustrate the computational details.

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.