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.     

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.     

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.     

Finite Population Variance Estimation in Presence of Measurement Errors

In this article, we propose a class of estimators for the population variance of a quantity of interest. The estimators in the class use auxiliary information to improve efficiency, and we suppose that measurement errors are present both in the study and auxiliary variate. We take into account such problem using a regression approach. We show that the class proposed is quite flexible and general, allowing to consider many kinds of information as auxiliary one. Comparisons within estimators in the class are studied theoretically and through simulations.     

Marginal hazard regression for correlated failure time data with auxiliary covariates

In many biomedical studies, it is common that due to budget constraints, the primary covariate is only collected in a randomly selected subset from the full study cohort. Often, there is an inexpensive auxiliary covariate for the primary exposure variable that is readily available for all the cohort subjects. Valid statistical methods that make use of the auxiliary information to improve study efficiency need to be developed. To this end, we develop an estimated partial likelihood approach for correlated failure time data with auxiliary information. We assume a marginal hazard model with common baseline hazard function. The asymptotic properties for the proposed estimators are developed. The proof of the asymptotic results for the proposed estimators is nontrivial since the moments used in estimating equation are not martingale-based and the classical martingale theory is not sufficient. Instead, our proofs rely on modern empirical process theory. The proposed estimator is evaluated through simulation studies and is shown to have increased efficiency compared to existing methods. The proposed method is illustrated with a data set from the Framingham study.     

An improved alternative estimation procedure for current population mean in presence of positively and negatively correlated auxiliary variables in two-occasion rotation patterns

Abstract

The present study confirms the influential role of a positively and a negatively correlated auxiliary variables in enhancing the precision of estimates of current population mean in two occasion rotation (successive) sampling. Exponential-type estimators of current population mean have been proposed for three different situations: (i) the information on a positively correlated auxiliary variable is readily available on both occasions (ii) the information on a negatively correlated auxiliary variable is readily available on both occasions and (iii) the information on both positively and negatively correlated auxiliary variables are readily available on both the occasions. The characteristics of the proposed estimators have been explored and their efficacious performances are compared with the natural and recent contemporary estimators. Optimum replacement strategies of the proposed estimation procedures have been formulated. Simulation and empirical studies are carried out to justify the proposition of the proposed estimators and appropriate recommendations have been put forward to the survey practitioners.     

Additive hazards regression of current status data with auxiliary covariates

This paper discusses the regression analysis of current status failure time data arising from the additive hazards model with auxiliary covariates. As often occurs in practice, it is impossible or impractical to measure the exact magnitude of covariates for all subjects in a study. To compensate the missing information, some auxiliary covariates are utilized instead. We propose two easy-to-implement procedures for estimation of regression parameters by making use of auxiliary information. The asymptotic properties of the resulting estimators are established and extensive numerical studies indicate that both procedures work well in practice.     

An Improved Class of Estimators for the General Population Parameters Using Auxiliary Information

This article suggests an improved class of estimators for estimating the general population parameter using information on an auxiliary variable. The properties of the suggested class of estimators have been studied under large sample approximation. The general results are then applied to estimate the population coefficient of variation of study variable using auxiliary information. An empirical study is given in support of the theoretical results.     

Efficient M-estimators with auxiliary information

This paper introduces a new class of M-estimators based on generalised empirical likelihood (GEL) estimation with some auxiliary information available in the sample. The resulting class of estimators is efficient in the sense that it achieves the same asymptotic lower bound as that of the efficient generalised method of moment (GMM) estimator with the same auxiliary information. The paper also shows that in case of smooth estimating equations the proposed estimators enjoy a small second order bias property compared to both efficient GMM and full GEL estimators. Analytical formulae to obtain bias corrected estimators are also provided. Simulations show that with correctly specified auxiliary information the proposed estimators and in particular those based on empirical likelihood outperform standard M and efficient GMM estimators both in terms of finite sample bias and efficiency. On the other hand with moderately misspecified auxiliary information estimators based on the nonparametric tilting method are typically characterised by the best finite sample properties.     

A class of chain ratio–product type estimators with two auxiliary variables under double sampling scheme

In this paper, a chain ratio–product type estimators has been developed for estimating population mean of the study variable using two auxiliary variables under double sampling scheme, when the information on another additional auxiliary character is available along with the main auxiliary character. The optimum property of the proposed strategy has been identified in two different cases with their variances. Theoretical and empirical studies have been done to demonstrate the efficiency of the proposed estimator over other estimators, which utilized the information on two auxiliary characteristics.     

Estimation of the distribution function with calibration methods

The paper proposes a new calibration estimator for the distribution function of the study variable. This estimator is a distribution function unlike others estimators that use auxiliary information. Comparisons are made with existing estimators in two simulation studies.     

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.     

Calibration for Randomized Response Estimators

In the present article, we consider the calibration procedure for the Warner's and Mangat–Singh's (:M–S) randomized response survey estimators using auxiliary information associated with the variable of interest. In the calibration procedure, we can use auxiliary information such as age, gender, and income for the respondents of RR questions from an external source, and then the classical RR estimators can be improved with respect to the problems of noncoverage or nonresponse. From the efficiency comparison study, we show that the calibration estimators are more efficient than those of Warner's and Mangat-Singh's when the known population cell and marginal counts of auxiliary information are used for the calibration procedure.     

Some estimation procedures using a linear model in successive sampling

ABSTRACT

In successive sampling some recent works depict the use of super-population models where information on stable auxiliary variable over occasions has been utilized. Stability character of auxiliary variable may not sustain, if the duration between occasions is large. To cope with such situations, the present work is an attempt to develop some estimation procedures by utilizing the information on two independent auxiliary variables through a linear super-population model. Some estimators are proposed to estimate the current population mean in two occasions successive (rotation) sampling. Optimum replacement strategies are formulated and performances of the proposed estimators have been discussed. Results are interpreted through empirical studies.     

Distribution estimation with auxiliary information for missing data

There is much literature on statistical inference for distribution under missing data, but surprisingly very little previous attention has been paid to missing data in the context of estimating distribution with auxiliary information. In this article, the auxiliary information with missing data is proposed. We use Zhou, Wan and Wang's method (2008) to mitigate the effects of missing data through a reformulation of the estimating equations, imputed through a semi-parametric procedure. Whence we can estimate distribution and the τth quantile of the distribution by taking auxiliary information into account. Asymptotic properties of the distribution estimator and corresponding sample quantile are derived and analyzed. The distribution estimators based on our method are found to significantly outperform the corresponding estimators without auxiliary information. Some simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.     

Using multiparametric auxiliary information at the estimation stage

Difference type estimators use auxiliary information based on an auxiliary parameter (specifically the parameter of interest), associated with the auxiliary variable. In practice, however, several parameters for auxiliary variables are available. This paper discusses how such estimators can be modified to improve the usual methods if information related to other parameters associated with an auxiliary variable or variables is available. Some applications estimating several such parameters are described. A proper set of simulation-based comparisons is made. Research partially supported by MCYT (Spain) contract n. BFM2001-3190     

Some Classes of Estimators for Median Estimation in Survey Sampling

In this article, we have suggested some classes of estimators for estimating finite population median using information on an auxiliary variable. To study the properties of suggested classes of estimators under large sample approximation, a generalized class of estimators has been suggested with its properties. It has been shown that the suggested classes of estimators are more efficient than other existing estimators. The results have been illustrated through an empirical study.     

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.     

A functional approach to estimating finite population distribution functions

Existing estimators of a finite population distribution function that utilize auxiliary information are often constructed by a point wise argument. As a result, these estimators are not always monotone. We adopt a functional approach to the problem and propose two estimators based on compositions of functions. Asymptotic variance formulae are derived for the proposed es-timators. Comparisons are made with existing estimators in a simulation study using three natural populations.     

Estimation of mean using double sampling for stratification and multivariate auxiliary information

Several estimators for estimating the mean of a principal variable are proposed based on double sampling for stratification (DSS) and multivariate auxiliary information. The general properties of the proposed estimators are studied, search for optimum estimators is made and the proposed estimators are compared with the corresponding estimators based on unstratified double sampling (USDS).