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 for small areas in household surveys

In many socio-economic surveys the objective is estimation of total or proportion of persons with a particular attribute. Multi-stage area samples are drawn from geographic strata and population within areal units is used as an auxiliary variable in ratio estimation. For large administrative areas, the auxiliary variable totals are available as population projections based on the last census. However, for small areas population changes are significantly affected by non-demographic factors and hence projections with high enough reliability are not available for small areas. In such situations the efficiency of design-based estimators for small areas can be improved by a ratio adjustment based on the auxiliary variable total for a large area. An inequality on the efficiency of the ratio adjusted estimator is established and its bias and variance is investigated     

Inference on Polychotomous Responses in Finite Populations

Abstract. A model‐based predictive estimator is proposed for the population proportions of a polychotomous response variable, based on a sample from the population and on auxiliary variables, whose values are known for the entire population. The responses for the non‐sample units are predicted using a multinomial logit model, which is a parametric function of the auxiliary variables. A bootstrap estimator is proposed for the variance of the predictive estimator, its consistency is proved and its small sample performance is compared with that of an analytical estimator. The proposed predictive estimator is compared with other available estimators, including model‐assisted ones, both in a simulation study involving different sampling designs and model mis‐specification, and using real data from an opinion survey. The results indicate that the prediction approach appears to use auxiliary information more efficiently than the model‐assisted approach.     

A Note on the Hartley-Ross Unbiased Ratio Estimator

In this article, we will construct an alternative Hartley-Ross unbiased ratio estimator for the population mean of a study variable when related auxiliary information is available. We will also present some efficiency comparisons and related results and give a numerical illustration.     

基于排序集样本和辅助变量中位数的均值比率估计方法

排序集抽样下利用辅助变量中位数构建了总体均值的改进比率估计模型,分析了该比率估计量的偏差和均方误差,并与简单随机抽样下的比率估计比较,证明了改进后的比率估计均方误差更小。以农作物播种面积和产量为研究对象进行实例分析,研究表明,基于排序集样本和辅助变量中位数的比率估计方法可以有效提高估计精度,验证了该构造方法的可行性。     

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.     

An efficiency comparison of dual ratio and product estimatiors

The present paper investigates the efficiency of the dual ratio estimator under the super population model with uncorrelated errors and a gamma-distributed auxiliary variabble. It is found that the dual ratio estimator is more efficient than the product estimator when the auxiliary variable has a gamma distribution with parameter greater than or equal to one, in the case when the regression is through the origin or when the product of intercept and slope is positive.     

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.     

Rare and clustered population estimation using the adaptive cluster sampling with some robust measures

The use of robust measures helps to increase the precision of the estimators, especially for the estimation of extremely skewed distributions. In this article, a generalized ratio estimator is proposed by using some robust measures with single auxiliary variable under the adaptive cluster sampling (ACS) design. We have incorporated tri-mean (TM), mid-range (MR) and Hodges-Lehman (HL) of the auxiliary variable as robust measures together with some conventional measures. The expressions of bias and mean square error (MSE) of the proposed generalized ratio estimator are derived. Two types of numerical study have been conducted using artificial clustered population and real data application to examine the performance of the proposed estimator over the usual mean per unit estimator under simple random sampling (SRS). Related results of the simulation study show that the proposed estimators provide better estimation results on both real and artificial population over the competing estimators.     

MODEL-ASSISTED HIGHER-ORDER CALIBRATION OF ESTIMATORS OF VARIANCE

In survey sampling, interest often centres on inference for the population total using information about an auxiliary variable. The variance of the estimator used plays a key role in such inference. This study develops a new set of higher‐order constraints for the calibration of estimators of variance for various estimators of the population total. The proposed strategy requires an appropriate model for describing the relationship between the response and auxiliary variable, and the variance of the auxiliary variable. It is therefore referred to as a model‐assisted approach. Several new estimators of variance, including the higher‐order calibration estimators of the variance of the ratio and regression estimators suggested by Singh, Horn & Yu and Sitter & Wu are special cases of the proposed technique. The paper presents and discusses the results of an empirical study to compare the performance of the proposed estimators and existing counterparts.     

Setting confidence intervals by ratio estimator

For the survey population total of a variable y when values of an auxiliary variable x are available a popular procedure is to employ the ratio estimator on drawing a simple random sample without replacement (SRSWOR) especially when the size of the sample is large. To set up a confidence interval for the total, various variance estimators are available to pair with the ratio estimator. We add a few more variance estimators studded with asymptotic design-cum-model properties. The ratio estimator is traditionally known to be appropriate when the regression of y on x is linear through the origin and the conditional variance of y given x is proportional to x. But through a numerical exercise by simulation we find the confidence intervals to fare better if the regression line deviates from the origin or if the conditional variance is disproportionate with x. Also, comparing the confidence intervals using alternative variance estimators we find our newly proposed variance estimators to yield favourably competitive results.     

Estimation of Mode Using Auxiliary Information

In this article, we propose naïve estimator, ratio estimator, and difference estimator of the mode of a study variable by using the known mode of an auxiliary variable. The asymptotic properties of the proposed estimators are studied analytically as well as empirically for different situations. The proper use of auxiliary information is found to result in efficient ratio and difference estimators than the naïve estimator.     

Chain-type ratio estimator in successive sampling using multi-auxiliary information

ABSTRACT

The present work is an attempt to make use of several auxiliary variables at both the occasions for improving the precision of estimates at the current occasion on two occasions of successive sampling. Chain-type ratio estimator has been proposed for estimating the population mean at current occasion in two occasions rotation (successive) sampling. Theoretical properties of the proposed estimator have been investigated. The proposed estimator has been compared with simple mean estimator when there is no matching and with the ratio estimator in successive sampling when information is available on one auxiliary variable on both the occasions. Optimum replacement strategy has also been discussed. Theoretical results have been justified through empirical investigation.     

Unbiased estimators of finite population variance using auxiliary information in sample surveys

We present some unbiased estimators of the population variance in a finite population sample survey using the knowledge of population variance of an auxiliary character.Exact variance expressions for the proposed estimators are obtained and compared with usual unbiased estimator and the ratio estimator envisaged by Isaki (1983). Generalization of the proposed estimator is also suggested.     

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.     

Estimation of rare and clustered population variance in adaptive cluster sampling

Abstract

Many researchers used auxiliary information together with survey variable to improve the efficiency of population parameters like mean, variance, total and proportion. Ratio and regression estimation are the most commonly used methods that utilized auxiliary information in different ways to get the maximum benefits in the form of high precision of the estimators. Thompson first introduced the concept of Adaptive cluster sampling, which is an appropriate technique for collecting the samples from rare and clustered populations. In this article, a generalized exponential type estimator is proposed and its properties have been studied for the estimation of rare and highly clustered population variance using single auxiliary information. A numerical study is carried out on a real and artificial population to judge the performance of the proposed estimator over the competing estimators. It is shown that the proposed generalized exponential type estimator is more efficient than the adaptive and non adaptive estimators under conventional sampling design.     

Calibration Approach-based Product Estimator of Finite Population Total with Subsampling of Nonrespondents under Single and Two-phase Sampling

Hansen and Hurwitz (1946) technique–based estimator of population total is proposed using the calibration approach under the assumption that the auxiliary variable is negatively correlated with the study variable. The variance estimation is also considered. The two-phase sampling case is also explored. The theoretical results are demonstrated through empirical studies using both generated and real population data. The proposed estimator of population total outperforms the existing estimators in terms of the criteria of relative bias and relative root mean square error.     

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.     

Improving ratio-type quantile estimates in a finite population

This paper presents a technique for improving the ratio method of estimation for finite population quantiles. The performance of this estimator with respect to others is studied theoretically and empirically, for a wide variety of real and artificial populations, and includes simple random sampling and sampling proportional to an auxiliary variable.