Dinesh K. Rao

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.

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.

Calibration estimator of population mean in stratified random sampling

Stratified Sampling is one of the most widely used sampling techniques as it increases the precision of the estimate of the survey variable. On the other hand, calibration estimation is a method of adjusting the original design weights to improve survey estimates by using auxiliary information such as the known population total (or mean) of the auxiliary variables. A calibration estimator uses calibrated weights that are determined to minimize a given distance measure to the original design weights while satisfying a set of constraints related to the auxiliary information. In this paper, a new calibration estimator of population mean in stratified sampling design is proposed, which incorporates not only the population mean but also the variance stratified mean available for the auxiliary variable. The problem of determining the optimum calibrated weights is formulated as a Nonlinear Programming Problem (NLPP) that is solved using Lagrange multiplier technique. The computational details of the procedure are illustrated in the presence of one auxiliary variable. A numerical example is presented and a simulation study is carried out to illustrate the computational details and the performance of the proposed calibration estimator. The results reveal that the proposed calibration estimator is more efficient than the other calibration estimators of the population mean.

Stratified calibration estimator of population mean using multi-auxiliary information

Calibration estimator uses calibrated weights that minimize a given distance measure to the design weights while satisfying a set of constraints based on the known auxiliary information such as population total (or mean). In this paper, we propose an improved multivariate calibration estimator of population mean in stratified sampling design. The problem of determining the calibrated weights is solved using Lagrange multiplier technique. The proposed multivariate calibration estimator of population mean is derived in the form of linear regression estimator (LREG). A numerical example is presented to illustrate the application and computational details of the proposed estimator.

A procedure for computing optimal stratum boundaries and sample sizes for multivariate surveys

In most surveys, the target variables (items of interest) commonly resemble right-skewed distributions where the Stratified Random Sampling technique is used as a method of sampling and estimation. The methodology of constructing strata is called stratification. Over a particular characteristic chosen as the stratification variable (such as gender, geographical region, ethnicity, or any natural criteria), the survey may fail to form homogeneous strata - this would impact the precision in the estimates of the target variables. Stratification can lead to substantial improvements in the precision of sample estimators, which not only depends on the sample size, but also on the heterogeneity among the units of the population. The principal reason for stratification in the design of sample surveys is to reduce the variance of sample estimates. Surveys normally have more than one target variable with several variables both available and desirable for stratification. Stratification in such multivariate situations has not been explored to a great deal like the univariate case and requires algorithms to determine efficient stratum boundaries. This paper takes into consideration multiple survey variables and attempts to present a computational procedure to construct optimal stratum boundaries (OSB) using Dynamic Programming (DP) technique. A numerical example to determine the OSB for two main variables under study is also presented.

Computing optimal stratum boundaries for multivariate surveys

IN most surveys, the target variables (items of interest) commonly resemble right-skewed distributions where the Stratified Random Sampling technique is used as a method of sampling and estimation. The methodology of constructing strata is called stratification. Over a particular characteristic chosen as the stratification variable (such as gender, geographical region, ethnicity, or any natural criteria), the survey may fail to form homogeneous strata-this would impact the precision in the estimates of the target variables. Stratification can lead to substantial improvements in the precision of sample estimators, which not only depends on the sample size, but also on the heterogeneity among the units of the population. The principal reason for stratification in the design of sample surveys is to reduce the variance of sample estimates. Surveys normally have more than one target variable with several variables both available and desirable for stratification. Stratification in such multivariate situations has not been explored to a great deal like the univariate case and requires algorithms to determine efficient stratum boundaries. This paper takes into consideration multiple survey variables and attempts to present a computational procedure to construct optimal stratum boundaries (OSB) using Dynamic Programming (DP) technique. A numerical example to determine the OSB for two main variables under study is also presented.