Fuzzy multi-objective optimization for optimum allocation in multivariate stratified sampling with quadratic cost and parabolic fuzzy numbers

This article deals with the uncertainties in a multivariate stratified sampling problem. The uncertain parameters of the problem, such as stratum standard deviations, measurement costs, travel costs and total budget of the survey, are considered as parabolic fuzzy numbers and the problem is formulated as a fuzzy multi-objective nonlinear programming problem with quadratic cost function. Using α-cut, parabolic fuzzy numbers are defuzzified and then the compromise allocations of the problem are obtained by fuzzy programming for a prescribed value of α. To demonstrate the utility of the proposed problem a numerical example is solved with the help of [LINGO User?s Guid. Lindo Systems Inc., 1415 North Dayton Street, Chicago,Illinois-60622, (USA), 2013] software and the derived compromise optimum allocation is compared with deterministic and proportional allocations.     

Compromise allocation in multivariate stratified surveys with stochastic quadratic cost function

In many real life situations the linear cost function does not approximate the actual cost incurred adequately. The cost of traveling between the units selected in the sample within a stratum is significant, instead of linear cost function. In this paper, we have considered the problem of finding a compromise allocation for a multivariate stratified sample survey with a significant travel cost within strata is formulated as a problem of non-linear stochastic programming with multiple objective functions. The compromise solutions are obtained through Chebyshev approximation technique, D ₁- distance and goal programming. A numerical example is presented to illustrate the computational details of the proposed methods.     

Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function

In multivariate stratified sample survey with L strata, let p-characteristics are defined on each unit of the population. To estimate the unknown p-population means of each characteristic, a random sample is taken out from the population. In multivariate stratified sample survey, the optimum allocation of any characteristic may not be optimum for others. Thus the problem arises to find out an allocation which may be optimum for all characteristics in some sense. Therefore a compromise criterion is needed to workout such allocation. In this paper, the procedure of estimation of p-population means is discussed in the presence of nonresponse when the use of linear cost function is not advisable. A solution procedure is suggested by using lexicographic goal programming problem. The numerical illustrations are given for its practical utility.     

Trade-off between cost and variance for a multi-objective compromise allocation in stratified random sampling

In the present paper, a multi-objective goal optimization mechanism is developed by trading off between cost and variance. Both are adversaries to each other while allocating a sample size even in stratified sampling design. Discussion section shows how these adversaries put their influence on optimal selection. This is a dual optimization procedure in which variance or mean square error is optimized in the first step and then considering some compromise on variance, cost is optimized. The process is applied to both individual and multi-objective programming models.     

Multi-objective nonlinear programming problem approach in multivariate stratified sample surveys in the case of non-response

In multivariate cases, usually the minimization of sampling variances is considered as an objective under a cost constraint. Since the variances are not unit free, it is more logical to consider the minimization of the squared coefficients of variation as an objective. In this paper, the problem of optimum compromise allocation in multivariate stratified sampling in the case of non-response as a multi-objective all-integer nonlinear programming problem is described. A solution procedure using four different approaches is considered, namely the value function, goal programming,∈-constraint and distance based, to obtain the compromise allocation for non-response. A numerical example is also presented to illustrate the computational details.     

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.     

Designing stratified sampling in economic and business surveys

In most economic and business surveys, the target variables (e.g. turnover of enterprises, income of households, etc.) commonly resemble skewed distributions with many small and few large units. In such surveys, if a stratified sampling technique is used as a method of sampling and estimation, the convenient way of stratification such as the use of demographical variables (e.g. gender, socioeconomic class, geographical region, religion, ethnicity, etc.) or other natural criteria, which is widely practiced in economic surveys, may fail to form homogeneous strata and is not much useful in order to increase the precision of the estimates of variables of interest. In this paper, a stratified sampling design for economic surveys based on auxiliary information has been developed, which can be used for constructing optimum stratification and determining optimum sample allocation to maximize the precision in estimate.     

On quantiles estimation based on different stratified sampling with optimal allocation

This work considers the problem of estimating a quantile function based on different stratified sampling mechanism. First, we develop an estimate for population quantiles based on stratified simple random sampling (SSRS) and extend the discussion for stratified ranked set sampling (SRSS). Furthermore, the asymptotic behavior of the proposed estimators are presented. In addition, we derive an analytical expression for the optimal allocation under both sampling schemes. Simulation studies are designed to examine the performance of the proposed estimators under varying distributional assumptions. The efficiency of the proposed estimates is further illustrated by analyzing a real data set from CHNS.     

The gini-simpson index of diversity: estimation in the stratified sampling

In previous papers the problem of estimating the Gini-Simpson index of diversity for large populations has been considered by using random samplings with and without replacement, Nevertheless, the populations to which this estimation is usually applied (e.g., anthropoiogicai, ecological, linguistic and sociological populations) often arise naturally stratified.

In this paper we first construct unbiased estimators of the Gini-Simpson index from a sample drawn according to a stratified sampling with proportional allocation and independently in different strata. Then, we determine the standard error of such estimators. The advantages of the stratification in estimating diversity are later confirmed by means of a practical example. We finally suggest complementary studies that could be additionally developed.     

On stratified bivariate ranked set sampling for regression estimators

We investigate the relative performance of stratified bivariate ranked set sampling (SBVRSS), with respect to stratified simple random sampling (SSRS) for estimating the population mean with regression methods. The mean and variance of the proposed estimators are derived with the mean being shown to be unbiased. We perform a simulation study to compare the relative efficiency of SBVRSS to SSRS under various data-generating scenarios. We also compare the two sampling schemes on a real data set from trauma victims in a hospital setting. The results of our simulation study and the real data illustration indicate that using SBVRSS for regression estimation provides more efficiency than SSRS in most cases.     

New calibration estimator in stratified sampling

This paper develops the theory of calibration estimation and proposes calibration approach alternative to existing calibration estimators for estimating population mean of the study variable using auxiliary variable in stratified sampling. The theory of new calibration estimation is given and optimum weights are derived. A simulation study is carried out to performance of the proposed calibration estimator with other existing calibration estimators. The results reveal that the proposed calibration estimators are more efficient than Tracy et al., Singh et al., Singh calibration estimators of the population mean.     

An adaptive allocation sampling design for which the conventional stratified estimator is an appropriate estimator

Adaptive allocations in stratified sampling design are suitable for studying Biological and Environmental populations. Biologists tend to use the conventional stratified estimator for an adaptive allocation sampling design in spite of its adaptive nature. In this paper, we have introduced an adaptive allocation sampling design for which the conventional stratified estimator has some desired statistical properties. We conduct a simulation study on a real population. The results show that the conventional stratified estimator for the introduced adaptive allocation sampling design is more efficient than other available estimators.     

Some calibration estimators for finite population mean in two-stage stratified random sampling

Abstract

In the present article, an effort has been made to develop calibration estimators of the population mean under two-stage stratified random sampling design when auxiliary information is available at primary stage unit (psu) level. The properties of the developed estimators are derived in-terms of design based approximate variance and approximate consistent design based estimator of the variance. Some simulation studies have been conducted to investigate the relative performance of calibration estimator over the usual estimator of the population mean without using auxiliary information in two-stage stratified random sampling. Proposed calibration estimators have outperformed the usual estimator without using auxiliary information.     

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 Cochran's average allocation and proportional allocation.     

Large sample results for tests of association ii. multinomial and stratified sampling

This paper deals with the asymptotics of a class of tests for association in 2-way contingency tables based on square forms in cell frequencies, given the total number of observations (multinomial sampling) or one set of marginal totals (stratified sampling). The case when both row and column marginal totals are fixed (hypergeometric sampling) was studied in Kulinskaya (1994), The class of tests under consideration includes a number of classical measures for association, Its two subclasses are the tests based on statistics using centralized cell frequencies (asymptotically distributed as weighted sums of central chi-squares) and those using the non-centralized cell frequencies (asymptotically normal). The parameters of asymptotic distributions depend on the sampling model and on true marginal probabilities. Maximum efficiency for asymptotically normal statistics is achieved under hypergeometric sampling, If the cell frequencies or the statistic as a whole are centralized using marginal proportions as estimates for marginal probabilities, the asymptotic distribution does not differ much between models and it is equivalent to that under hypergeometric sampling. These findings give an extra justification for the use of permutation tests for association (which are based on hypergeometric sampling). As an application, several well known measures of association are analysed.     

Estimation of a finite population total in two-stage and stratified sampling under superpopulation models

Optimal sampling strategies which minimise the expected mean square error for a linear design as well as model-design unbiased estimators for a finite population total for two-stage and stratified sampling are obtained under different superpopu1ation models     

On improvement in estimating population mean in stratified random sampling

Gupta and Shabbir 2 Gupta, S. and Shabbir, J. 2008. On improvement in estimating the population mean in simple random sampling. J. Appl. Stat., 35(5): 559–566. [Taylor & Francis Online], [Web of Science ®] [Google Scholar] have suggested an alternative form of ratio-type estimators for estimating the population mean. In this paper, we obtained a corrected version for the mean square error (MSE) of the Gupta–Shabbir estimator, up to first order of approximation, and the optimum case is discussed. We expand this estimator to the stratified random sampling and propose general classes for combined and separate estimators. Also an empirical study is carried out to show the properties of the proposed estimators.     

A generalized class of estimators under two-phase stratified sampling for non response

In this paper, we propose a generalized class of estimators for finite population mean using two auxiliary variables in two-phase stratified sampling for non response. We identify 17 estimators as special cases of the proposed class of estimators. Expressions for the bias and mean squared error (MSE) of estimators are obtained up to first order of approximation. A data set is used for efficiency comparisons.