Compromise mixed allocation using fuzzy programming approach

In sample surveys, stratified sampling is useful if the strata weights are known for each stratum. If they are not known double sampling may be used by selecting a large preliminary sample to estimate the strata weights. Then a stratified sample may be selected independently or from the initial sample. If the problem of non-response is also present then the strata are to be virtually divided into two disjoint and exhaustive groups of respondents and non-respondents. A subsamples from non-respondents is then selected and second more extensive attempt is to be group is to be made for obtaining the required information. The problem of obtaining a compromise allocation for first and second phase of sampling is the formulated as a multi-objective non- linear programming problem that minimizes the sum of variances of the stratified sample mean subject to the non-linear cost constraint. The formulated problem is solved using fuzzy programming and fuzzy goal programming based on piecewise linear approximation. A numerical example is presented to illustrate the computational details. Keywords: Non-response, Multivariate Stratified Sampling, Multiobjective Integer non-linear programming problem, Fuzzy Programming, Travel cost

In this paper, we propose a calibration estimator of population mean in stratified sampling using the known mean and variance information from multi-auxiliary variables. The problem of determining the optimum calibrated weights is formulated as a Nonlinear Programming Problem (NLPP) that is solved using the Lagrange multiplier technique. Numerical example with real data is presented to illustrate the computational details of the proposed estimator. A comparison study is also carried out using real and simulated data to evaluate the performance and the usefulness of the proposed estimator. The study reveals that the proposed estimator with multi-auxiliary information is more efficient estimator of the population mean as it provides least estimated variance and highest gain in relative efficiency (RE). References Jean Claude Deville and Carl Erik Sarndal. Calibration estimators in survey sampling. Journal of the American statistical Association, 87(418):376–382, 1992. doi:10.1080/01621459.1992.10475217 . Victor M Estevao and Carl Erik Sarndal. Survey estimates by calibration on complex auxiliary information. International Statistical Review, 74(2):127–147, 2006. doi:110.1111/j.1751-5823.2006.tb00165.x Patrick J Farrell and Sarjinder Singh. Model-assisted higher-order calibration of estimators of variance. Australian and New Zealand Journal of Statistics, 47(3):375–383, 2005. doi:10.1111/j.1467-842X.2005.00402.x Wolfram Research, Inc. Mathematica, Version 11.3. Champaign, IL, 2018. Jong Min Kim, Engin A Sungur, and Tae Young Heo. Calibration approach estimators in stratified sampling. Statistics and probability letters, 77(1):99–103, 2007. doi:10.1016/j.spl.2006.05.015 Phillip S Kott. Using calibration weighting to adjust for nonresponse and coverage errors. Survey Methodology, 32(2):133, 2006. Dinesh K Rao. Mathematical programing in stratified random sampling. PhD thesis, School of Computing, Information and Mathematical Sciences, The University of the South Pacific, Fiji, February 2017. Dinesh K. Rao, Tokaua. Tekabu, and Mohammad G M Khan. New calibration estimators in stratified sampling. In Proceedings of Asia-Pacific World Congress on Computer Science and Engineering, pages 66–70. IEEE, 2016. Gurmindar K Singh, Dinesh K Rao, and Mohammed GM Khan. Calibration estimator of population mean in stratified random sampling. In Proceedings of Asia-Pacific World Congress on Computer Science and Engineering (APWC on CSE), pages 1–5. IEEE, 2014. Sarjindar Singh, Stephen Horn, and Frank Yu. Estimation of variance of general regression estimator: Higher level calibration approach. Survey Methodology, 24:41–50, 1998. Sarjinder Singh. Advanced Sampling Theory With Applications: How Michael Selected Amy, volume I and II. Kluwer Academic Publishers, Netherlands, 2003. Sarjinder Singh. On the calibration of design weights using a displacement function. Metrika, 75(1):85–107, 2012. doi:10.1007/s00184-010-0316-6 Sarjinder Singh, Stephen Horn, Sadeq Chowdhury, and Frank Yu. Theory and methods: Calibration of the estimators of variance. Australian and New Zealand Journal of Statistics, 41(2):199–212, 1999. doi:10.1111/1467-842X.00074 D S Tracy, S Singh, and R Arnab. Note on calibration in stratified and double sampling. Survey Methodology, 29(1):99–104, 2003. Changbao Wu and Randy R Sitter. A model-calibration approach to using complete auxiliary information from survey data. Journal of the American Statistical Association, 96(453):185–193, 2001. doi:10.1198/016214501750333054

In this article, a combined general family of estimators is proposed for estimating finite population mean of a sensitive variable in stratified random sampling with non-sensitive auxiliary variable based on randomized response technique. Under stratified random sampling without replacement scheme, the expression of bias and mean square error (MSE) up to the first-order approximations are derived. Theoretical and empirical results through a simulation study show that the proposed class of estimators is more efficient than the existing estimators, i.e., usual stratified random sample mean estimator, Sousa et al (2014) ratio and regression estimator of the sensitive variable in stratified sampling.

Adapting a global stratified random sample for regional estimation of forest cover change derived from satellite imagery

In this study, comparison has been made for different sampling designs, using the HIES data of North West Frontier Province (NWFP) for 2001-02 and 1998-99 collected from the Federal Bureau of Statistics, Statistical Division, Government of Pakistan, Islamabad. The performance of the estimators has also been considered using bootstrap and Jacknife. A two-stage stratified random sample design is adopted by HIES. In the first stage, enumeration blocks and villages are treated as the first stage Primary Sampling Units (PSU). The sample PSU’s are selected with probability proportional to size. Secondary Sampling Units (SSU) i.e., households are selected by systematic sampling with a random start. They have used a single study variable. We have compared the HIES technique with some other designs, which are: Stratified Simple Random Sampling. Stratified Systematic Sampling. Stratified Ranked Set Sampling. Stratified Two Phase Sampling. Ratio and Regression methods were applied with two study variables, which are: Income (y) and Household sizes (x). Jacknife and Bootstrap are used for variance replication. Simple Random Sampling with sample size (462 to 561) gave moderate variances both by Jacknife and Bootstrap. By applying Systematic Sampling, we received moderate variance with sample size (467). In Jacknife with Systematic Sampling, we obtained variance of regression estimator greater than that of ratio estimator for a sample size (467 to 631). At a sample size (952) variance of ratio estimator gets greater than that of regression estimator. The most efficient design comes out to be Ranked set sampling compared with other designs. The Ranked set sampling with jackknife and bootstrap, gives minimum variance even with the smallest sample size (467). Two Phase sampling gave poor performance. Multi-stage sampling applied by HIES gave large variances especially if used with a single study variable.

Since 1967, when the Polish State Forests were first inventoried, a continuous increase in the volume of growing stock has been recorded. This increase in timber resources is mainly the result of sustainable forest management. However, during this time period inventory methods have been changed a few times and this may have affected the estimates of of wood resources. Since 2011 new instructions have been in place for forest management plans in the Polish State Forests. Despite this, the method of taking forest inventories remained consistent with the previous guidelines. This should allow us to consider changes in the volume of growing stock in relation to inventory methods and their accuracy. In this paper, the results of growing stock estimations based on two assessments made using stratified random sampling are compared. After five years of inventories made for forest management plans, 422 sample plots in two forest districts were measured within 15 strata. Predicted age classes structure at the beginning of successive management plans was used to determine new strata. Data from two inventories were compared on the basis of age class. In addition, data from the State Forests Information System i.e. updated stand level inventory data (on felling and tree growth) were analyzed. Comparing data from the two inventories, in three of seven age classes there was a decrease in the volume of growing stock after a five year period. We found no rational explanation for this. The extent of felling in the analyzed stand was very low, as confirmed by data from State Forests database. However, the largest decrease in volume was 1.5%, whereas the accuracy of growing stock assessment of discussed age class was estimated at some 9% in 2006 and 8% in 2011. Hence, from statistical viewpoint there was no differences between the results of both inventories. However, the differences between volume of growing stock of examined age classes, updated in the Information System of the State Forests and results of assessments by stratified sampling are less than 95% confidence intervals for means of samples. Because of that, updated information could be used in the long term management planning. The results clearly suggest that changes of growing stock, estimated as a difference between two assessments, should be interpreted in conjunction with the accuracy of these inventories. The accuracy of sampling should be also taken into account when creating management plans.

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.

Kappa coefficient is a popular statistic to measure the agreement between the classifications of two raters. The estimator of the coefficient and its variance are approximately satisfied by a simple random sampling method. To get more efficient results for confidence interval estimation of kappa, stratified random sampling method can also be used, alternatively. In this study, a bootstrap method under stratified sampling is suggested to use to estimate the confidence interval of kappa coefficient. The results are discussed over three data sets.

This paper presents the results of a limited investigation which brings into focus the difficulties encountered in developing exact distribution-free methods for stratified simple random sampling. It is shown that exact distribution-free confidence intervals for a population median can be obtained from a stratified sample under very special circumstances. The principal result concerns the order statistics of the combined separate strata samples, where proportional allocation is employed. It is proved that a pair of symmetric order statistics provides a confidence interval for the population median whose confidence coefficient is not less than the confidence coefficient obtained from the corresponding order statistics of a random sample of the same total size drawn from the entire population. Furthermore, it is shown by means of a numerical example that this result necessarily holds only when proportional allocation is employed.

In multivariate sampling, the major interest is on the problem of estimation of several population characteristics which often make conflicting demands on the sampling procedure. In this type of survey, the best allocation for one item may not in general be the best for another. There is the need to come up with compromise solution in a survey with many characteristics under study. This paper focuses on comparing some techniques of optimum sample allocation which are Yates/Chatterjee, Booth and Sedransk and Vector Maximum Criterion (VMC) on five sets of real life data stratified into six strata and two variates with desired variances using: (i.) Method of maximum variances with fixed n and (ii.) arbitrary fixing of variances. The stratum sample size nh among the classes are obtained to ascertain the criterion that will produce the smallest n. Based on the set of data collected and used for the empirical study it was discovered that Vector Maximum Criterion (VMC), Booth and Sedransk are superior to Yates/Chatterjee.

Measurement errors may significantly distort the properties of an estimator. In this paper, estimators of the finite population variance using the information on first and second raw moments of the study variable are developed under stratified random sampling that incorporate the variance of a measurement error component. Additionally, combined and separate estimators are also developed for estimating the finite population variance using supplementary information in terms of one or two auxiliary variables. An empirical study is carried out to study the effect of measurement error on the relative efficiencies of the proposed estimators.

Fast integer-valued algorithms for optimal allocations under constraints in stratified sampling

We consider here the problem of allocating sampling effort among strata when there is no useful information on the within-stratum variability. We show that in a natural game theoretic analogue proportional allocation is a unique good strategy.

Survey non-response refers to the failure to get a useable response from the respondent selected for the sample. Non-response may occur due to several reasons. The randomised response technique (RRT) presented by Warner (1965) used to estimate proportion of respondents to the sensitive questions without revealing his choice to the interviewer. In this article, the problem of optimal allocation in stratified sampling where RRT is used in presence of non-response, is formulated as an nonlinear programming problem (NLPP). The formulated problem is solved using branch and bound method and the results are obtained through LINGO.

Considering the possible correlation between the characteristics (variables) in multivariate stratified random sampling, a modified Prekopa’s approach is suggested for the problem of optimum allocation in multivariate stratified random sampling. An example is solved by applying the proposed methodology.