A simulation study: estimation of population mean using two auxiliary variables in stratified random sampling

ABSTRACT

This paper deals with the problem of estimating the finite population mean in stratified random sampling by using two auxiliary variables. This paper proposed a ratio-cum-product exponential type estimator of population mean under different situations: (i) when there is presence of non-response and measurement errors on the study as well as auxiliary variables; (ii) when there is non-response on the study and auxiliary variables but with no measurement error; (iii) when there is complete response on study variable but there is presence of non-response and measurement error on the auxiliary variables and (iv) when there are complete response and measurement error on study as well as auxiliary variables. The expressions of the bias and mean square error of the proposed estimator have been obtained up to the first degree of approximation. The proposed estimator has been compared with usual unbiased estimator, ratio estimator and other existing estimators and the conditions obtained to show the efficacy of the proposed estimator over other considered estimators. Simulation study is carried out to support the theoretical findings.     

A regression estimator for finite population mean of a sensitive variable using an optional randomized response model

Kalucha et al. (Kalucha G., Gupta S., Dass B. K. (accepted). Ratio estimation of finite population mean using optional randomized response models. Journal of Statistical Theory and Practice) introduced an additive ratio estimator for finite population mean of a sensitive variable in simple random sampling without replacement and showed that this estimator performs better than the ordinary mean estimator based on an optional randomized response technique (RRT). In this paper, we introduce a regression estimator that performs better than the ratio estimator even for the modest correlation between the study and the auxiliary variables. A comparison of the proposed estimator with the corresponding ratio estimator and the ordinary RRT mean estimator is carried out theoretically, and is also illustrated with a simulation study.     

A Class of Estimators of Population Variance Using Auxiliary Information in Sample Surveys

This article advocates the problem of estimating the population variance of the study variable using information on certain known parameters of an auxiliary variable. A class of estimators for population variance using information on an auxiliary variable has been defined. In addition to many estimators, usual unbiased estimator, Isaki's (1983), Upadhyaya and Singh's (1999), and Kadilar and Cingi's (2006) estimators are shown as members of the proposed class of estimators. Asymptotic expressions for bias and mean square error of the proposed class of estimators have been obtained. An empirical study has been carried out to judge the performance of the various estimators of population variance generated from the proposed class of estimators over usual unbiased estimator, Isaki's (1983), Upadhyaya and Singh's (1999) and Kadilar and Cingi's (2006) estimators.     

Use of several auxiliary variables in estimating the population mean in a two-occasion rotation pattern

ABSTRACT

This paper addresses the problem of estimation of the population mean on the current (second) occasion in two-occasion successive sampling. Utilizing the readily available information on several auxiliary variables on both occasions and the information on the study variable from the previous occasion, an estimation procedure of the population mean on the current occasion has been proposed. Theoretical properties of the proposed estimator have been investigated. Optimum replacement policy to the proposed estimator has been discussed. The proposed estimator has been compared empirically with the sample mean estimator, when there is no matching and the optimum estimator which is a linear combination of the means of the matched and unmatched portions of the sample at the current occasion. Appropriate recommendations have been made for practical applications.     

Estimation of the Mean of a Sensitive Variable in the Presence of Auxiliary Information

Sousa et al. (2010 Sousa , R. , Shabbir , J. , Real , P. C. , Gupta , S. ( 2010 ). Ratio estimation of the mean of a sensitive variable in the presence of auxiliary information . J. Statist. Theor. Prac. 4 ( 3 ): 495 – 507 .[Taylor & Francis Online] , [Google Scholar]) introduced a ratio estimator for the mean of a sensitive variable and showed that this estimator performs better than the ordinary mean estimator based on a randomized response technique (RRT). In this article, we introduce a regression estimator that performs better than the ratio estimator even for modest correlation between the primary and the auxiliary variables. The underlying assumption is that the primary variable is sensitive in nature but a non sensitive auxiliary variable exists that is positively correlated with the primary variable. Expressions for the Bias and MSE (Mean Square Error) are derived based on the first order of approximation. It is shown that the proposed regression estimator performs better than the ratio estimator and the ordinary RRT mean estimator (that does not utilize the auxiliary information). We also consider a generalized regression-cum-ratio estimator that has even smaller MSE. An extensive simulation study is presented to evaluate the performances of the proposed estimators in relation to other estimators in the study. The procedure is also applied to some financial data: purchase orders (a sensitive variable) and gross turnover (a non sensitive variable) in 2009 for a population of 5,336 companies in Portugal from a survey on Information and Communication Technologies (ICT) usage.     

Exponential-Type Estimators of the Mean of a Sensitive Variable in the Presence of Nonsensitive Auxiliary Information

Sousa et al. and Gupta et al. suggested ratio and regression-type estimators of the mean of a sensitive variable using nonsensitive auxiliary variable. This article proposes exponential-type estimators using one and two auxiliary variables to improve the efficiency of mean estimator based on a randomized response technique. The expressions for the mean squared errors (MSEs) and bias, up to first-order approximation, have been obtained. It is shown that the proposed exponential-type estimators are more efficient than the existing estimators. The gain in efficiency over the existing estimators has also been shown with a simulation study and by using real data.     

Estimation of population mean based on dual use of auxiliary information in non response

Whenever there is auxiliary information available in any form, the researchers want to utilize it in the method of estimation to obtain the most efficient estimator. When there exists enough amount of correlation between the study and the auxiliary variables, and parallel to these associations, the ranks of the auxiliary variables are also correlated with the study variable, which can be used a valuable device for enhancing the precision of an estimator accordingly. This article addresses the problem of estimating the finite population mean that utilizes the complementary information in the presence of (i) the auxiliary variable and (ii) the ranks of the auxiliary variable for non response. We suggest an improved estimator for estimating the finite population mean using the auxiliary information in the presence of non response. Expressions for bias and mean squared error of considered estimators are derived up to the first order of approximation. The performance of estimators is compared theoretically and numerically. A numerical study is carried out to evaluate the performances of estimators. It is observed that the proposed estimator is more efficient than the usual sample mean and the regression estimators, and some other families of ratio and exponential type of estimators.     

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.     

Change of origin and scale in ratio and difference methods of estimation in sampling

This paper discusses two estimators of the mean of a finite population based on a simple random sample from it, when supplementary information on a variable positively correlated with the variable of interest is available. Simultaneous reductions in absolute bias and mean square error of the estimator are seen as compared with those of the traditional estimator in the ratio method of estimation. The suggested estimators are simple for computation and there is no appreciable increase in the cost as well.     

A Generalized Class of Ratio Type Exponential Estimators of Population Mean Under Linear Transformation of Auxiliary Variable

Recently, Shabbir and Gupta [Shabbir, J. and Gupta, S. (2011). On estimating finite population mean in simple and stratified random sampling. Communications in Statistics-Theory and Methods, 40(2), 199–212] defined a class of ratio type exponential estimators of population mean under a very specific linear transformation of auxiliary variable. In the present article, we propose a generalized class of ratio type exponential estimators of population mean in simple random sampling under a very general linear transformation of auxiliary variable. Shabbir and Gupta's [Shabbir, J. and Gupta, S. (2011). On estimating finite population mean in simple and stratified random sampling. Communications in Statistics-Theory and Methods, 40(2), 199–212] class of estimators is a particular member of our proposed class of estimators. It has been found that the optimal estimator of our proposed generalized class of estimators is always more efficient than almost all the existing estimators defined under the same situations. Moreover, in comparison to a few existing estimators, our proposed estimator becomes more efficient under some simple conditions. Theoretical results obtained in the article have been verified by taking a numerical illustration. Finally, a simulation study has been carried out to see the relative performance of our proposed estimator with respect to some existing estimators which are less efficient under certain conditions as compared to the proposed estimator.     

Efficient utilization of two auxiliary variables in stratified double sampling

In this article, we propose a new difference-type estimator in estimating the finite population mean in stratified double sampling by using the ranks of two auxiliary variables as an additional information. The proposed estimator performs better than the usual sample mean estimator, ratio estimator, exponential estimator, Choudhury and Singh (2012) estimator, Vishwakarma and Gangele (2014) estimator, Singh and Khalid (2015) estimator, Khan and Al-Hossain (2016) estimator, Khan (2016) estimator, and the usual difference estimator. Two real datasets are used to observe the performances of estimators.     

Estimation of population mean in two-occasion successive sampling

In the present article, we have proposed some classes of estimators based on transformed auxiliary variable. The biases and mean squared errors (MSEs) of the proposed estimators have been obtained. The proposed estimators have been compared with simple mean estimator when there is no matching and the optimum estimator, which is a combination of the means of the matched and unmatched portion of the sample at the second occasion. Optimum replacement policy and the efficiency of the proposed estimators have been discussed. Theoretical results are well supported with an empirical study.     

Preliminary Test Regression Estimator When two Prior Values of Population Mean (μx) of an Auxiliary Variable (x) are Available

A regression estimator using two prior values of population mean (μ_x) of an auxiliary variable (x) is proposed after a preliminary test of closeness of these prior values to the true valueμ_x. The proposed preliminary test regression estimator has been found to be more efficient in general than the usual regression estimator when prior values are used in place of μ_xwithout preliminary test of significance. The efficiency of the proposed estimator over the usual regression estimator has also been computed for different values of Δ₀, Δ₁, n, and ρ, which showed considerable gain in precision.     

Generalized estimator of population mean by using conventional and non-conventional measures in the presence of measurement errors

In survey research, it is assumed that reported response by the individual is correct. However, given the issues of prestige bias, self-respect, respondent's reported data often produces estimated values which are highly deviated from the true values. This causes measurement error (ME) to be present in the sample estimates. In this article, the estimation of population mean in the presence of measurement error using information on a single auxiliary variable is studied. A generalized estimator of population mean is proposed. The class of estimators is obtained by using some conventional and non-conventional measures. Simulation and numerical study is also conducted to assess the performance of estimators in the presence and absence of measurement error.     

A modified bootstrap estimator for the mean of an asymmetric distribution

A modified bootstrap estimator of the population mean is proposed which is a convex combination of the sample mean and sample median, where the weights are random quantities. The estimator is shown to be strongly consistent and asymptotically normally distributed. The small- and moderate-sample-size behavior of the estimator is investigated and compared with that of the sample mean by means of Monte Carlo studies. It is found that the newly proposed estimator has much smaller mean squared errors and also yields significantly shorter confidence intervals for the population mean.     

Calibrated Estimators of Population Mean for a Mail Survey Design

In this article, we consider the problem of estimating the population mean of a study variable in the presence of non-response in a mail survey design. We introduce calibrated estimators of the population mean of a study variable in the presence of a known auxiliary variable. Using simulation the proposed calibrated estimators of population mean are compared to the Hansen and Hurwitz (1946) estimator under different situations for fixed cost as well for fixed sample size. The results are then extended for the use of multi-auxiliary information and stratified random sampling. We consider the problem of estimating the average total family income in the US in the presence of known auxiliary information on total income per person, age of the person, and poverty. We compute the relative efficiency of the proposed estimator over the Hansen and Hurwitz (1946) estimator through the use of large real datasets. Results are also presented for sub-populations consisting of whites, blacks, others, and two or more races in addition to considering them together in a population.     

A new method of imputation in survey sampling

In this paper, an alternative estimator of population mean in the presence of non-response has been suggested which comes in the form of Walsh's estimator. The estimator of mean obtained from the proposed technique remains better than the estimators obtained from ratio or mean methods of imputation. The mean-squared error (MSE) of the resultant estimator is less than that of the estimator obtained on the basis of ratio method of imputation for the optimum choice of parameters. An estimator for estimating a parameter involved in the process of new method of imputation has been discussed. A suggestion to form ‘warm deck’ method of imputation has been suggested. The MSE expressions for the proposed estimators have been derived analytically and compared empirically. The work has been extended to the case of multi-auxiliary information to be used for imputation. Numerical illustrations are also provided.     

Estimation of population mean in presence of random non response in two-stage cluster sampling

The present investigation addresses the problem of estimating a finite population mean in two-phase cluster sampling in presence of random non response situations. Utilizing information on an auxiliary variable, regression type estimators has been proposed. Effective imputation techniques have been suggested to deal with the random non response situations. The properties of the proposed estimation strategies have been studied for different cases of random non response situations in practical surveys. The superiority of the suggested methodology over the natural sample mean estimator of population mean has been established through empirical studies carried over the data sets of natural population and artificially generated population.     

Sampling design proportional to order statistic of auxiliary variable

The sampling designs dependent on sample moments of auxiliary variables are well known. Lahiri (Bull Int Stat Inst 33:133–140, 1951) considered a sampling design proportionate to a sample mean of an auxiliary variable. Sing and Srivastava (Biometrika 67(1):205–209, 1980) proposed the sampling design proportionate to a sample variance while Wywiał (J Indian Stat Assoc 37:73–87, 1999) a sampling design proportionate to a sample generalized variance of auxiliary variables. Some other sampling designs dependent on moments of an auxiliary variable were considered e.g. in Wywiał (Some contributions to multivariate methods in, survey sampling. Katowice University of Economics, Katowice, 2003a); Stat Transit 4(5):779–798, 2000) where accuracy of some sampling strategies were compared, too.These sampling designs cannot be useful in the case when there are some censored observations of the auxiliary variable. Moreover, they can be much too sensitive to outliers observations. In these cases the sampling design proportionate to the order statistic of an auxiliary variable can be more useful. That is why such an unequal probability sampling design is proposed here. Its particular cases as well as its conditional version are considered, too. The sampling scheme implementing this sampling design is proposed. The inclusion probabilities of the first and second orders were evaluated. The well known Horvitz–Thompson estimator is taken into account. A ratio estimator dependent on an order statistic is constructed. It is similar to the well known ratio estimator based on the population and sample means. Moreover, it is an unbiased estimator of the population mean when the sample is drawn according to the proposed sampling design dependent on the appropriate order statistic.