Bias-adjustment and calibration of jackknife variance estimator in the presence of non-response

In this paper, bias-adjustment in the jackknife estimator of variance accredited to Rao and Sitter (1995) has been considered. Then the bias-adjusted Rao and Sitter (1995) estimator has been calibrated such that its expected value under the imputing superpopulation model remains the same as the expected value of the mean squared error of the ratio estimator in the presence of non-response. A simulation study has been performed to compare the six different estimators of variance: out of them four estimators belong to Rao and Sitter (1995) and the other two proposed estimators are named as bias-adjusted and bias-adjusted-cum-calibrated estimators. The empirical relative bias and empirical relative efficiency of the two proposed estimators with respect to the four existing estimators accredited to Rao and Sitter (1995) have been investigated through simulations. The bias-adjusted-cum-calibrated estimator has been found to be an efficient estimator in the case of heteroscadastic populations. The present paper considers the situation of simple random and without replacement sampling. The possibility of obtaining a negative estimate of variance by the estimator due to Kim et al. (2006) has been pointed out.     

General class of estimators in multi-character surveys

In this paper, a general class of estimators for the estimation of a finite population total in multi-character surveys is proposed. It is shown that the estimators proposed by Arnab (2002), Amahiaet al. (1989) and Bansal and Singh (1985) are the special cases of the proposed class of estimators. The proposed class of estimators is always more efficient than the estimator proposed by Rao (1966).     

Randomized Response Techniques Using Maximum Likelihood Estimator

The randomized response (RR) technique with two decks of cards proposed by Odumade and Singh (2009) can always be made more efficient than the RR techniques proposed by Warner (1965), Mangat and Singh (1990), and Mangat (1994) by adjusting the proportion of cards in the decks. Arnab et al. (2012) generalized Odumade and Singh strategy (2009) for complex survey designs and wider class of estimators. In this paper improvement of Arnab et al. (2012) estimator has been made by using maximum likelihood method.     

Mass imputation for two-phase sampling

Two-phase sampling is a cost-effective method of data collection using outcome-dependent sampling for the second-phase sample. In order to make efficient use of auxiliary information and to improve domain estimation, mass imputation can be used in two-phase sampling. Rao and Sitter (1995) introduce mass imputation for two-phase sampling and its variance estimation under simple random sampling in both phases. In this paper, we extend the Rao–Sitter method to general sampling design. The proposed method is further extended to mass imputation for categorical data. A limited simulation study is performed to examine the performance of the proposed methods.     

Mean squared error estimators of small area means using survey weights

Using survey weights, You & Rao [You and Rao, The Canadian Journal of Statistics 2002; 30, 431–439] proposed a pseudo‐empirical best linear unbiased prediction (pseudo‐EBLUP) estimator of a small area mean under a nested error linear regression model. This estimator borrows strength across areas through a linking model, and makes use of survey weights to ensure design consistency and preserve benchmarking property in the sense that the estimators add up to a reliable direct estimator of the mean of a large area covering the small areas. In this article, a second‐order approximation to the mean squared error (MSE) of the pseudo‐EBLUP estimator of a small area mean is derived. Using this approximation, an estimator of MSE that is nearly unbiased is derived; the MSE estimator of You & Rao [You and Rao, The Canadian Journal of Statistics 2002; 30, 431–439] ignored cross‐product terms in the MSE and hence it is biased. Empirical results on the performance of the proposed MSE estimator are also presented. The Canadian Journal of Statistics 38: 598–608; 2010 © 2010 Statistical Society of Canada     

Small-area estimation by combining time-series and cross-sectional data

A model involving autocorrelated random effects and sampling errors is proposed for small-area estimation, using both time-series and cross-sectional data. The sampling errors are assumed to have a known block-diagonal covariance matrix. This model is an extension of a well-known model, due to Fay and Herriot (1979), for cross-sectional data. A two-stage estimator of a small-area mean for the current period is obtained under the proposed model with known autocorrelation, by first deriving the best linear unbiased prediction estimator assuming known variance components, and then replacing them with their consistent estimators. Extending the approach of Prasad and Rao (1986, 1990) for the Fay-Herriot model, an estimator of mean squared error (MSE) of the two-stage estimator, correct to a second-order approximation for a small or moderate number of time points, T, and a large number of small areas, m, is obtained. The case of unknown autocorrelation is also considered. Limited simulation results on the efficiency of two-stage estimators and the accuracy of the proposed estimator of MSE are présentés.     

Performance of the empirical Bayes estimator for fixed parameters

When the unbiased estimators of a set of parameters are independently and normally distributed, the Empirical Bayes Estimator (EB) for each of the parameters depends on all the parameters. When these parameters are considered to be fixed, Rao and Shinozaki (1978) [7] compared the mean square error (MSE) of this estimator for an individual parameter with the variance of its unbiased estimator, and cautioned that its bias may be large. In this article, the conditions required for (a) the MSE of the EB to be smaller than the variance of the unbiased estimator and (b) at the same time, for its bias to be smaller than a specified fraction of the square root of the MSE are evaluated. To satisfy these conditions, critical limits for the difference of the parameter from the average of all the parameters and the sum of such differences over all the parameters are determined. As an illustration, for the daily inpatient hospital expenses in the Metropolitan Statistical Areas (MSAs) of 15 states in the US, the sample means and EBs are compared through the estimates of these limits.     

Meta-analysis of Proportions of Rare Events–A Comparison of Exact Likelihood Methods with Robust Variance Estimation

The conventional random effects model for meta-analysis of proportions approximates within-study variation using a normal distribution. Due to potential approximation bias, particularly for the estimation of rare events such as some adverse drug reactions, the conventional method is considered inferior to the exact methods based on binomial distributions. In this article, we compare two existing exact approaches—beta binomial (B-B) and normal-binomial (N-B)—through an extensive simulation study with focus on the case of rare events that are commonly encountered in medical research. In addition, we implement the empirical (“sandwich”) estimator of variance into the two models to improve the robustness of the statistical inferences. To our knowledge, it is the first such application of sandwich estimator of variance to meta-analysis of proportions. The simulation study shows that the B-B approach tends to have substantially smaller bias and mean squared error than N-B for rare events with occurrences under 5%, while N-B outperforms B-B for relatively common events. Use of the sandwich estimator of variance improves the precision of estimation for both models. We illustrate the two approaches by applying them to two published meta-analysis from the fields of orthopedic surgery and prevention of adverse drug reactions.     

Variance estimation for the finite population distribution function with complete auxiliary information

The authors develop jackknife and analytical variance estimators for the estimator of Chambers & Dunstan (1986) and Rao, Kovar & Mantel (1990) of the finite population distribution function, using complete auxiliary information. They also describe the associated model and show the design consistency of the variance estimators, whose small‐sample performance is examined through a limited simulation study. They highlight the operational advantages of the jackknife in the model‐based setting of Chambers & Dunstan (1986) and its better conditional performance in the design‐based setting of Rao, Kovar & Mantel (1990).     

A family of estimators of population variance in two-occasion rotation patterns

Abstract

In this article, we have considered the problem of estimation of population variance on current (second) occasion in two occasion successive (rotation) sampling. A class of estimators of population variance has been proposed and its asymptotic properties have been discussed. The proposed class of estimators is compared with the sample variance estimator when there is no matching from the previous occasion and the Singh et al. (2013) estimator. Optimum replacement policy is discussed. It has been shown that the suggested estimator is more efficient than the Singh et al. (2013) estimator and a usual unbiased estimator when there is no matching. An empirical study is carried out in support of the present study.     

A pseudo‐empirical best linear unbiased prediction approach to small area estimation using survey weights

The authors develop a small area estimation method using a nested error linear regression model and survey weights. In particular, they propose a pseudo‐empirical best linear unbiased prediction (pseudo‐EBLUP) estimator to estimate small area means. This estimator borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. The proposed estimator also has a nice self‐benchmarking property. The authors also obtain an approximation to the model mean squared error (MSE) of the proposed estimator and a nearly unbiased estimator of MSE. Finally, they compare the proposed estimator with the EBLUP estimator and the pseudo‐EBLUP estimator proposed by Prasad & Rao (1999), using data analyzed earlier by Battese, Harter & Fuller (1988).     

Testing equality of ratios from two finite populations

Approximations are given for the bias and variance of both the regression and ratio estimator when sampling from a finite population, and simulation results are given indicating the accuracy of the approximations and the bias of the estimated approximations. A different estimator for the variance of the regression estimator is recommended. Test procedures are proposed for testing the hypothesis of equality of ratios from two finite populations, the procedures depending upon the types of populations being sampled. Simulation results indicating the effectiveness of the test procedures in controlling their size are given.     

Efficiency of projected score methods in rectangular array asymptotics

Summary. The paper considers a rectangular array asymptotic embedding for multistratum data sets, in which both the number of strata and the number of within-stratum replications increase, and at the same rate. It is shown that under this embedding the maximum likelihood estimator is consistent but not efficient owing to a non-zero mean in its asymptotic normal distribution. By using a projection operator on the score function, an adjusted maximum likelihood estimator can be obtained that is asymptotically unbiased and has a variance that attains the Cramér–Rao lower bound. The adjusted maximum likelihood estimator can be viewed as an approximation to the conditional maximum likelihood estimator.     

An Improved Generalized Difference-Cum-Ratio-Type Estimator for the Population Variance in Two-Phase Sampling Using Two Auxiliary Variables

In this paper, an improved generalized difference-cum-ratio-type estimator for the finite population variance under two-phase sampling design is proposed. The expressions for bias and mean square error (MSE) are derived to first order of approximation. The proposed estimator is more efficient than the usual sample variance estimator, traditional ratio estimator, traditional regression estimator, chain ratio type and chain ratio-product-type estimators, and Jhajj and Walia (2011) estimator. Four datasets are also used to illustrate the performances of different estimators.     

Fully efficient estimation of coefficients of correlation in the presence of imputed survey data

Marginal imputation, that consists of imputing items separately, generally leads to biased estimators of bivariate parameters such as finite population coefficients of correlation. To overcome this problem, two main approaches have been considered in the literature: the first consists of using customary imputation methods such as random hot‐deck imputation and adjusting for the bias at the estimation stage. This approach was studied in Skinner & Rao 2002 . In this paper, we extend the results of Skinner & Rao 2002 to the case of arbitrary sampling designs and three variants of random hot‐deck imputation. The second approach consists of using an imputation method, which preserves the relationship between variables. Shao & Wang 2002 proposed a joint random regression imputation procedure that succeeds in preserving the relationships between two study variables. One drawback of the Shao–Wang procedure is that it suffers from an additional variability (called the imputation variance) due to the random selection of residuals, resulting in potentially inefficient estimators. Following Chauvet, Deville, & Haziza 2011 , we propose a fully efficient version of the Shao–Wang procedure that preserves the relationship between two study variables, while virtually eliminating the imputation variance. Results of a simulation study support our findings. An application using data from the Workplace and Employees Survey is also presented. The Canadian Journal of Statistics 40: 124–149; 2012 © 2011 Statistical Society of Canada     

Calibration approach estimation of the mean in stratified sampling and stratified double sampling

Calibration estimation improves the precision of the estimates of population parameters by incorporating specified auxiliary information. A class of calibration estimators has been proposed for estimating the population mean by making use of a set of calibration constraints in stratified sampling. The estimator of variance of the proposed calibration estimator of the mean is derived using a lower level calibration approach. The idea is extended for stratified double sampling. A simulation study is used to evaluate the performances of the proposed estimators by comparing them with the similar estimators developed by Tracy, Singh and Arnab (2003 Tracy, D.S., Singh, S., Arnab, R. (2003). Note on calibration in stratified and double sampling. Surv. Methodol. 29(1): 99–104. [Google Scholar]) based on different sets of calibration constraints.     

On the restricted almost unbiased estimators in linear regression

In this paper, the restricted almost unbiased ridge regression estimator and restricted almost unbiased Liu estimator are introduced for the vector of parameters in a multiple linear regression model with linear restrictions. The bias, variance matrices and mean square error (MSE) of the proposed estimators are derived and compared. It is shown that the proposed estimators will have smaller quadratic bias but larger variance than the corresponding competitors in literatures. However, they will respectively outperform the latter according to the MSE criterion under certain conditions. Finally, a simulation study and a numerical example are given to illustrate some of the theoretical results.     

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.     

Bias Corrected Maximum Likelihood Estimator Under the Generalized Linear Model for a Binary Variable

Under the generalized linear models for a binary variable, an approximate bias of the maximum likelihood estimator of the coefficient, that is a special case of linear parameter in Cordeiro and McCullagh (1991), is derived without a calculation of the third-order derivative of the log likelihood function. Using the obtained approximate bias of the maximum likelihood estimator, a bias-corrected maximum likelihood estimator is defined. Through a simulation study, we show that the bias-corrected maximum likelihood estimator and its variance estimator have a better performance than the maximum likelihood estimator and its variance estimator.     

Estimation of Bowley's Coefficient of Skewness in the Presence of Auxiliary Information

In this paper, we suggest regression-type estimators for estimating the Bowley's coefficient of skewness using auxiliary information. To the first degree of approximation, the bias and mean-squared error expressions of the regression-type estimators are obtained, and the regions under which these estimators are more efficient than the conventional estimator are also determined. Further, a general class of estimators of the Bowley's coefficient of skewness is defined along with its properties. A class of estimators based on estimated optimum values is also defined. It is shown to the first degree of approximations that the variance of the class of estimators based on estimated optimum values is the same as that of the minimum variance of the proposed class of estimators. A simulation study is carried out to demonstrate the performance of the proposed difference estimator over the usual estimator.