Some remarks on the traditional way of circular and diagonal circular systematic sampling schemes

ABSTRACT

In this paper, some deficiencies in traditional selection procedure of circular version of systematic sampling schemes are investigated and alternative methods are proposed. We also suggest some rules of thumb for coincidence of units in the sample. The end corrections proposed by Bellhouse and Rao (1975 Bellhouse, D.R., Rao, J.N.K. (1975). Systematic sampling in the presence of a trend. Biometrika. 62:694–697.[Crossref], [Web of Science ®] , [Google Scholar]) and Sampath and Varalakshmi (2008) for circular systematic sampling and diagonal circular systematic sampling, respectively, are also modified.     

Estimators for the Population Mean in the Case of Missing Data

We propose a class of estimators for the population mean when there are missing data in the data set. Obtaining the mean square error equations of the proposed estimators, we show the conditions where the proposed estimators are more efficient than the sample mean, ratio-type estimators, and the estimators in Singh and Horn (2000 Singh , S. , Horn , S. ( 2000 ). Compromised imputation in survey sampling . Metrika 51 : 267 – 276 .[Crossref], [Web of Science ®] , [Google Scholar]) and Singh and Deo (2003 Singh , S. , Deo , B. (2003). Imputation by power transformation. Statist. Pap. 44:555–579.[Crossref], [Web of Science ®] , [Google Scholar]) in the case of missing data. These conditions are also supported by a numerical example.     

Star-Type Systematic Sampling Schemes for Finite Populations

Two new sampling schemes namely, Star-Type Systematic (STS) sampling without replacement and Modified Star-Type Systematic (MSTS) sampling without replacement for estimation of finite population means are introduced. The relative performances of the proposed star-type systematic sample means along with those of the simple random and systematic sample means are assessed for a hypothetical population with a linear trend and also for certain natural populations. Furthermore, the usefulness of the proposed sampling schemes in quality control and for constructing partial diallel crosses in mating designs are briefly break discussed.     

Sequential Estimation of Expectations in the Presence of Trend2

A sequence of independent random variables {Z_n:n≥ 1} with unknown probability distributions is considered and the problem of estimating their expectations {M_n+1: n≥ 1} is examined. The estimation of M_n+1 is based on a finite set {z_k:1≤k≤n}, each z_k being an observed value of Z_k, 1 ≤k≤n, and also based on the assumption that {M_n:n≥ 1} follows an unknown trend of a specified form.     

On the Use of Superpopulation Model in Estimation of Population Mean in Two-Occasion Rotation Patterns

The present work is an attempt to study the estimation of the population mean on the current occasion in two-occasion successive (rotation) sampling under a superpopulation model. Six different estimators are proposed for estimating the current population mean in two-occasion successive (rotation) sampling. Optimum replacement policies and performances of the proposed estimators have been discussed. Results are interpreted via empirical studies.     

Estimation of Finite Population Mean in Stratified Random Sampling with Two Auxiliary Variables under Double Sampling Design

In this article, a chain ratio-product type exponential estimator is proposed for estimating finite population mean in stratified random sampling with two auxiliary variables under double sampling design. Theoretical and empirical results show that the proposed estimator is more efficient than the existing estimators, i.e., usual stratified random sample mean estimator, Chand (1975) chain ratio estimator, Choudhary and Singh (2012) estimator, chain ratio-product-type estimator, Sahoo et al. (1993) difference type estimator, and Kiregyera (1984) regression-type estimator. Two data sets are used to illustrate the performances of different estimators.     

On the Evaluation of the Kolmogorov Statistic

At least two computer program packages, SPSS and STRATA, use simulated Bernoulli trials to draw (without replacement) a random sample of records from a finite population of records. Therefore, the size of the sample is a random variable. Two estimators of a population total under this sampling procedure are compared with the usual estimator under simple random sampling. Conditions under which the Bernoulli sampling estimators have almost the same mean squared error as the simple random-sample estimator are illustrated.     

Confidence Intervals for the Median of a Finite Population Based on the Sign Test

This article presents methods for constructing confidence intervals for the median of a finite population under simple random sampling without replacement, stratified random sampling, and cluster sampling. The confidence intervals, as well as point estimates and test statistics, are derived from sign estimating functions which are based on the well-known sign test. Therefore, a unified approach for inference about the median of a finite population is given.     

Unbiased Estimation of the Distribution Function of an Exponential Population Using Order Statistics with Application in Ranked Set Sampling

In this paper we consider the problem of unbiased estimation of the distribution function of an exponential population using order statistics based on a random sample. We present a (unique) unbiased estimator based on a single, say ith, order statistic and study some properties of the estimator for i = 2. We also indicate how this estimator can be utilized to obtain unbiased estimators when a few selected order statistics are available as well as when the sample is selected following an alternative sampling procedure known as ranked set sampling. It is further proved that for a ranked set sample of size two, the proposed estimator is uniformly better than the conventional nonparametric unbiased estimator, further, for a general sample size, a modified ranked set sampling procedure provides an unbiased estimator uniformly better than the conventional nonparametric unbiased estimator based on the usual ranked set sampling procedure.     

Improvement in Estimating the Population Median in Simple Random Sampling and Stratified Random Sampling Using Auxiliary Information

This paper deals with estimation of population median in simple and stratified random samplings by using auxiliary information. Auxiliary information is rarely used in estimating population median, although there have been many studies to estimate population mean using auxiliary information. In this study, we suggest some estimators using auxiliary information such as mode and range of an auxiliary variable and correlation coefficient. We also expand these estimators to stratified random sampling for combined and separate estimators. We obtain mean square error equations for all proposed estimators and find theoretical conditions. These conditions are also supported by using numerical examples.     

Evaluation of Linear Trend Tests Using Resampling Techniques

A number of parametric and non-parametric linear trend tests for time series are evaluated in terms of test size and power, using also resampling techniques to form the empirical distribution of the test statistics under the null hypothesis of no linear trend. For resampling, both bootstrap and surrogate data are considered. Monte Carlo simulations were done for several types of residuals (uncorrelated and correlated with normal and nonnormal distributions) and a range of small magnitudes of the trend coefficient. In particular for AR(1) and ARMA(1, 1) residual processes, we investigate the discrimination of strong autocorrelation from linear trend with respect to the sample size. The correct test size is obtained for larger data sizes as autocorrelation increases and only when a randomization test that accounts for autocorrelation is used. The overall results show that the type I and II errors of the trend tests are reduced with the use of resampled data. Following the guidelines suggested by the simulation results, we could find significant linear trend in the data of land air temperature and sea surface temperature.     

Estimation of Population Mean Under Non Response in Two-Occasion Rotation Patterns

The present article is an attempt to study the effect of non response at both occasions in search of good rotation patterns over two occasions. Ratio-type estimators were proposed for estimating the population mean at current occasion in presence of non response at both the occasions in two-occasion successive (rotation) sampling. Detail behaviors of proposed estimators were studied. Proposed estimators were compared with the estimator using no information from previous (first) occasion. Performances of the proposed estimators were demonstrated via empirical studies.     

Some Effective Rotation Patterns in Estimation of Population Mean in Two-Occasion Successive Sampling

This article is an attempt to explore some effective rotation patterns in estimation of current population mean in two occasions successive sampling. Utilizing the readily available information on an auxiliary variable on both occasions and the information on study variable from the previous occasion, some efficient estimation procedures have been suggested. Optimum replacement strategies and the efficiencies of the proposed estimators have been discussed. Empirical studies are carried out and suitable recommendations are made.     

Remainder modified systematic sampling in the presence of linear trend

If the population size is not a multiple of the sample size, then the usual linear systematic sampling design is unattractive, since the sample size obtained will either vary, or be constant and different to the required sample size. Only a few modified systematic sampling designs are known to deal with this problem and in the presence of linear trend, most of these designs do not provide favorable results. In this paper, a modified systematic sampling design, known as remainder modified systematic sampling (RMSS), is introduced. There are seven cases of RMSS and the results in this paper suggest that the proposed design is favorable, regardless of each case, while providing linear trend-free sampling results for three of the seven cases. To obtain linear trend-free sampling for the other cases and thus improve results, an end corrections estimator is constructed.     

Estimation of Finite Population Variance Using Scrambled Responses in the Presence of Auxiliary Information

In this article, a new estimator for estimating the finite population variance of a sensitive variable based on scrambled responses collected using a randomization device is introduced. The estimator is then improved by using known auxiliary information. The estimators due to Das and Tripathi (1978: Sankhya) and Isaki (1983: JASA) are shown to be special cases of the proposed estimator. Numerical simulations are performed to study the magnitude of the gain in efficiency when using the estimator with auxiliary information with respect to the estimator based only on the scrambled responses. An idea to extend the present work from SRSWOR design to more complex design is also given.     

Some Classes of Estimators for the Population Median Using Auxiliary Information

This article considers some classes of estimators of the population median of the study variable using information on an auxiliary variable with their properties under large sample approximation. Asymptotic optimum estimator (AOE) in each class of estimators has been investigated along with the approximate mean square error formulae. It has been shown that the proposed classes of estimators are better than these considered by Gross (1980 Gross , T. S. ( 1980 ). Median estimation in sample surveys. Proc. Surv. Res. Meth. Sect. Amer. Statist. Assoc. 181–184 . [Google Scholar]), Kuk and Mak (1989 Kuk , A. Y. C. , Mak , T. K. ( 1989 ). Median estimation in the presence of auxiliary information . J. Roy. Statist. Soc. Ser. B51 : 261 – 269 . [Google Scholar]), Singh et al. (2003a Singh , H. P. , Singh , S. , Joarder , A. H. ( 2003a ). Estimation of population median when mode of an auxiliary variable is known . J. Statist. Res. 37 ( 1 ): 57 – 63 . [Google Scholar]), and Al and Cingi (2009 Al , S. , Cingi , H. ( 2009 ). New estimators for the population median in simple random sampling. Tenth Islamic Countries Conference on Statistical Sciences, held in New Cairo, Egypt . [Google Scholar]). An empirical study is carried out to judge the merits of the suggested class of estimators over other existing estimators.     

An Efficient Class of Estimators for the Population Mean Using Auxiliary Information in Stratified Random Sampling

This article addresses the problem of estimating the population mean in stratified random sampling using the information of an auxiliary variable. A class of estimators for population mean is defined with its properties under large sample approximation. In particular, various classes of estimators are identified as particular member of the suggested class. It has been shown that the proposed class of estimators is better than usual unbiased estimator, usual combined ratio estimator, usual product estimator, usual regression estimator and Koyuncu and Kadilar (2009 Koyuncu, N., Kadilar, C. (2009). Ratio and product estimators in stratified random sampling. J. Statist. Plan. Infere. 139:2552–2558.[Crossref], [Web of Science ®] , [Google Scholar]) class of estimators. The results have been illustrated through an empirical study.     

Family of Estimators of Population Mean Using Two Auxiliary Variables in Stratified Random Sampling

A general family of estimators, which use the information of two auxiliary variables in the stratified random sampling, is proposed to estimate the population mean of the variable under study. Under stratified random sampling without replacement scheme, the expressions of bias and mean square error (MSE) up to the first- and second-order approximations are derived. The family of estimators in its optimum case is discussed. Also, an empirical study is carried out to show the properties of the proposed estimators.     

Ratio-Cum-Product Type Exponential Estimator of Finite Population Mean in Stratified Random Sampling

This article addresses the problem of estimating the finite population mean in stratified random sampling using auxiliary information. Motivated by Singh (1967 Singh , M. P. ( 1967 ). Ratio cum product method of estimation . Metrika 12 : 34 – 42 .[Crossref] , [Google Scholar]) and Bahl and Tuteja (1991 Bahl , S. , Tuteja , R. K. ( 1991 ). Ratio and product type exponential estimator . Inform. Optimiz. Sci. 12 ( 1 ): 159 – 163 .[Taylor &; Francis Online] , [Google Scholar]) a ratio-cum-product type exponential estimator has been suggested and its bias and mean squared error have been derived under large sample approximation. Suggested estimator has been compared with usual unbiased estimator of population mean in stratified random sampling, combined ratio estimator, combined product estimator, ratio and product type exponential estimator of Singh et al. (2008 Singh , R. , Kumar , M. , Singh , R. D. , Chaudhary , M. K. ( 2008 ). Exponential ratio type estimators in stratified random sampling. Presented in International Symposium on Optimisation and Statistics (I.S.O.S) at A.M.U., Aligarh, India, during 29–31 Dec . [Google Scholar]). Conditions under which suggested estimator is more efficient than other considered estimators have been obtained. A numerical illustration is given in support of the theoretical findings.     

Edge Estimation in the Population of a Binary Tree Using Node-Sampling

Suppose a finite population of several vertices, each connected to single or multiple edges. This constitutes a structure of graphical population of vertices and edges. As a special case, the graphical population like a binary tree having only two child vertices associated to parent vertex is taken into consideration. The entire binary tree is divided into two sub-graphs such as a group of left-nodes and a group of right-nodes. This paper takes into account a mixture of graph structured and population sampling theory together and presents a methodology for mean-edge-length estimation of left sub-graph using right edge sub-graph as an auxiliary source of information. A node-sampling procedure is developed for this purpose and a class of estimators is proposed containing several good estimators. Mathematical conditions for minimum bias and optimum mean squared error of the class are derived and theoretical results are numerically supported with a test of 99% confidence intervals. It is shown that suggested class has a sub-class of optimum estimators, and sample-based estimates are closer to the true value of the population parameter.