A NOTE ON COMPETING VARIANCE ESTIMATORS IN RANDOMISED RESPONSE SURVEYS

When gathering randomised rather than direct responses on a variable of interest relating to sensitive issues, one may use a modified version of the well-known generalised regression predictor of a finite population total. To construct confidence intervals, this paper proposes four alternative variance estimators – modifications to those usable with direct responses – and examines their relative efficiencies through simulations from simple super-population models.     

A NOTE ON VARIANCE ESTIMATION FOR THE GENERALIZED REGRESSION PREDICTOR

The generalized regression (GREG) predictor is used for estimating a finite population total when the study variable is well‐related to the auxiliary variable. In 1997, Chaudhuri & Roy provided an optimal estimator for the variance of the GREG predictor within a class of non‐homogeneous quadratic estimators (H) under a certain superpopulation model M. They also found an inequality concerning the expected variances of the estimators of the variance of the GREG predictor belonging to the class H under the model M. This paper shows that the derivation of the optimal estimator and relevant inequality, presented by Chaudhuri & Roy, are incorrect.     

VARIANCE ESTIMATION IN MULTI-STAGE SAMPLING

In multi-stage sampling with the first stage units (fsu) chosen without replacement (WOR) with varying probability schemes (VPS) unbiased estimators (UE) of variances of homogeneous linear (HL) functions of unbiased estimators (UE) T_i's of fsu totals Y_i's based on selection of subsequent stage units (SSU) from chosen fsu's are derived as homogeneous quadratic (HQ) functions of alternative less efficient UE's, say of T_i';'s of Y_i's. Specific strategies are illustrated.     

Calibration of the estimators of variance

This investigation suggests new techniques to calibrate estimators of variance. Estimators of the variance of simple mean, ratio and regression estimators under different sampling schemes are shown to be special cases of the proposed calibration techniques. The approach has more practical use due to recent advances in programming techniques and computational speed. An empirical study has been carried out to address the properties of these proposed strategies.     

EXAMPLES OF MINIMUM VARIANCE ESTIMATION1

In this sequel to a previous discussion of minimum variance estimation (Bartlett, 1982), the gain with conditional estimation procedures is illustrated for the location parameter for (i) the rectangular distribution; (ii) a triangular distribution (typifying an asymmetric case). This note concludes with further remarks on the multi-sample and multi-parameter cases.     

A Monte Carlo comparison of some variance estimators of the Horvitz–Thompson estimator

At the design and estimation stage of a survey, large survey organization often uses auxiliary information. This article discusses various procedures for improving variance estimation of the Horvitz–Thompson estimator of a finite population total with the aid of auxiliary information. To study the design-based properties of the proposed variance estimators relative to the standard one, a small scale Monte Carlo study is performed.     

COMPONENTS OF VARIANCE MODELS WITH TRANSFORMATIONS

Power transformations are a popular way to improve the agreement between the observations and the assumptions in a statistical model. In this paper it is assumed that the data, after appropriate power transformation Λ, satisfies a variance components model, with independent Gaussian components. The focus is on inference for quantities which have an interpretation regardless of the choice of Λ (Carroll & Ruppert, 1981) – in particular the intraclass correlation coefficient ρ, the predicted probability of a new observation being less than a specified value and the predicted quantile. It is shown that, in the case Λ= 0, the asymptotic variance of ρ is the same, whether or not one treats Δ as estimated or as known. This supports an empirical conjecture of Solomon (1985). For predicted probabilities and predicted quantiles the variance when A is estimated is shown to be only slightly greater than the variance assuming Δ is known, except in the tails of the distribution where there can be substantial difference between the two variances.     

A class of estimators using auxiliary information in sample surveys

A large class of estimators is considered for the mean of a finite population using information on an auxiliary variable. It is shown that members of this class of estimators are asymptotically no more efficient than the linear regression estimator.     

COMBINING TWO UNBIASED ESTIMATORS OF A COMMON MEAN OF TWO NORMAL POPULATIONS

Consider estimating the common mean μ of two normal populations. Let () and () be the means and variances of two independent samples obtained from these populations. We give sufficient conditions for the choices of α₁ and α₂ in the unbiased estimator     

A RESTRICTED MAXIMUM LIKELIHOOD PROCEDURE FOR ESTIMATING THE VARIANCE FUNCTION OF AN IMMUNOASSAY

Restricted maximum likelihood (REML) is a procedure for estimating a variance function in a heteroscedastic linear model. Although REML has been extended to non-linear models, the case in which the data are dominated by replicated observations with unknown values of the independent variable of interest, such as the concentration of a substance in a blood sample, has not been considered. We derive a REML procedure for an immunoassay and show that the resulting estimator is superior to those currently being used. Some interesting properties of the REML estimator are derived, and its relationship to other estimators is discussed.     

VARIANCE ESTIMATION IN THE ERROR COMPONENTS REGRESSION MODEL

ABSTRACT

In a regression model with a random individual and a random time effect explicit representations of the nonnegative quadratic minimum biased estimators of the corresponding variances are deduced. These estimators always exist and are unique. Moreover, under normality assumption of the dependent variable unbiased estimators of the mean squared errors of the variance estimates are derived. Finally, confidence intervals on the variance components are considered.     

Calibrated estimators in two-stage sampling

We consider the problem of the estimation of the population mean of a study variable by assuming that the population means of an auxiliary variable are known at both stages of sample selection. The design weights at the first and second stages of sample selection are calibrated by optimizing the chi-squared type distance between the design weights and the new weights at both the first and second stages of sample selection. The regression type estimator in two-stage sampling is shown to be a special case. An application of the proposed estimators using a real data set is discussed.     

KERNEL ESTIMATORS OF PROBABILITY DENSITY FUNCTIONS BY RANKED-SET SAMPLING

ABSTRACT

Kernel estimation of probability density functions is considered when ranked-set samples are available. The properties of the resulting estimators are derived for small and large samples, while performance with respect to the usual simple random sample estimators is investigated for a range of probability density models.     

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     

ON THE ROBUST ANALYSIS OF VARIANCE COMPONENTS MODELS FOR PEDIGREE DATA

Quantitative traits measured over pedigrees of individuals may be analysed using maximum likelihood estimation, assuming that the trait has a multivariate normal distribution. This approach is often used in the analysis of mixed linear models. In this paper a robust version of the log likelihood for multivariate normal data is used to construct M-estimators which are resistant to contamination by outliers. The robust estimators are found using a minimisation routine which retains the flexible parameterisations of the multivariate normal approach. Asymptotic properties of the estimators are derived, computation of the estimates and their use in outlier detection tests are discussed, and a small simulation study is conducted.     

Variance estimation in adaptive cluster sampling

Abstract

In this paper, a class of variance estimator is proposed of a finite population variance under an adaptive cluster sampling design in the presence of information on an auxiliary variable. We obtain expressions for the mean square error and bias for the developed estimators and their performance is evaluated on a Poisson clustered process and a real data set. The simulation study evaluates the efficiency of the suggested estimators for an adaptive cluster sampling (ACS) design and the Isaki (1983 Isaki, C. T. 1983. Variance estimation using auxiliary information. Journal of the American Statistical Association 78 (381):117–23. doi: 10.1080/01621459.1983.10477939.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) estimator of the variance for SRSWOR over the sample variance for SRSWOR.     

Two-step calibration of design weights under two auxiliary variables in sample survey

Calibration on the available auxiliary variables is widely used to increase the precision of the estimates of parameters. Singh and Sedory [Two-step calibration of design weights in survey sampling. Commun Stat Theory Methods. 2016;45(12):3510–3523.] considered the problem of calibration of design weights under two-step for single auxiliary variable. For a given sample, design weights and calibrated weights are set proportional to each other, in the first step. While, in the second step, the value of proportionality constant is determined on the basis of objectives of individual investigator/user for, for example, to get minimum mean squared error or reduction of bias. In this paper, we have suggested to use two auxiliary variables for two-step calibration of the design weights and compared the results with single auxiliary variable for different sample sizes based on simulated and real-life data set. The simulated and real-life application results show that two-auxiliary variables based two-step calibration estimator outperforms the estimator under single auxiliary variable in terms of minimum mean squared error.     

抽样调查中的校准估计法

本文梳理总结了校准估计法自首次提出以来的研究成果。理论方法的发展集中在最短距离法、工具向量法和模型校准法三种校准方法的研究上;方法应用的发展体现在对简单参数和复杂参数的校准估计上。对小域估计、无回答、二重抽样等特定抽样问题和总体分位数、总体方差估计中校准估计法的具体应用作了重点梳理介绍。对校准估计法的理论和应用研究前景作了展望。     

Median Estimation Using Double Sampling

This paper proposes a general class of estimators for estimating the median in double sampling. The position estimator, stratification estimator and regression type estimator attain the minimum variance of the general class of estimators. The optimum values of the first-phase and second-phase sample sizes are also obtained for the fixed cost and the fixed variance cases. An empirical study examines the performance of the double sampling strategies for median estimation. Finally, an extension of the methods of Chen & Qin (1993) and Kuk & Mak (1994) is considered for the double sampling strategy.