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.     

The generalized family of estimators of population mean using auxiliary information in double sampling

Abstract

We suggested the class of estimators of the population mean with its bias and mean square error. It has been shown that the suggested class is more efficient than the usual unbiased, ratio, product and regression estimators and estimators due to Bahl and Tuteja (1991), Singh et al. (2009), and Upadhyaya et al. (2011). In addition an empirical study also carried out to and founded that the members of suggested family also have improvement over Grover and Kaur (2011) and Shabbir and Gupta (2011) classes. Two-phase (double) sampling version of the proposed class was also given.     

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.     

Estimation of finite population variance in successive sampling using multi-auxiliary variables

ABSTRACT

In this paper, a general class of estimators for estimating the finite population variance in successive sampling on two occasions using multi-auxiliary variables has been proposed. The expression of variance has also been derived. Further, it has been shown that the proposed general class of estimators is more efficient than the usual variance estimator and the class of variance estimators proposed by Singh et al. (2011) when we used more than one auxiliary variable. In addition, we support this with the aid of numerical illustration.     

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.     

Estimation of population proportion of a qualitative character using randomized response technique in stratified random sampling

This paper proposes an efficient stratified randomized response model based on Chang et al.'s (2004) model. We have obtained the variance of the proposed estimator of π_s, the proportion of the respondents in the population belonging to a sensitive group, under proportional and Neyman allocations. It is shown that the estimator based on the proposed model is more efficient than the Chang et al.'s (2004) estimator under both proportional as well as Neyman allocations, Hong et al.'s (1994) estimator and Kim and Warde's (2004) estimator. Numerical illustration and pictorial representation are given in support of the present study.     

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.     

A stratified estimation of a sensitive character by using geometric distribution

Abstract

In this paper, we consider the estimation of a sensitive character when the population is consisted of several strata; this is undertaken by applying Niharika et al.’s model which is using geometric distribution as a randomization device. A sensitive parameter is estimated for the case in which stratum size is known, and proportional and optimum allocation methods are taken into account. We extended the Niharika et al.’s model to the case of an unknown stratum size; a sensitive parameter is estimated by applying stratified double sampling to the Niharika et al.’s model. Finally, the efficiency of the proposed model is compared with that of Niharika et al. in terms of the estimator variance.     

Modified ratio estimators using robust regression methods

When there is an outlier in the data set, the efficiency of traditional methods decreases. In order to solve this problem, Kadilar et al. (2007) adapted Huber-M method which is only one of robust regression methods to ratio-type estimators and decreased the effect of outlier problem. In this study, new ratio-type estimators are proposed by considering Tukey-M, Hampel M, Huber MM, LTS, LMS and LAD robust methods based on the Kadilar et al. (2007). Theoretically, we obtain the mean square error (MSE) for these estimators. We compared with MSE values of proposed estimators and MSE values of estimators based on Huber-M and OLS methods. As a result of these comparisons, we observed that our proposed estimators give more efficient results than both Huber M approach which was proposed by Kadilar et al. (2007) and OLS approach. Also, under all conditions, all of the other proposed estimators except Lad method are more efficient than robust estimators proposed by Kadilar et al. (2007). And, these theoretical results are supported with the aid of a numerical example and simulation by basing on data that includes an outlier.     

Doubly robust empirical likelihood inference in covariate-missing data problems

Missing covariate data occurs often in regression analysis. We study methods for estimating the regression coefficients in an assumed conditional mean function when some covariates are completely observed but other covariates are missing for some subjects. We adopt the semiparametric perspective of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Amer Statist Assoc. 1994;89:846–866] on regression analyses with missing covariates, in which they pioneered the use of two working models, the working propensity score model and the working conditional score model. A recent approach to missing covariate data analysis is the empirical likelihood method of Qin et al. [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503], which effectively combines unbiased estimating equations. In this paper, we consider an alternative likelihood approach based on the full likelihood of the observed data. This full likelihood-based method enables us to generate estimators for the vector of the regression coefficients that are (a) asymptotically equivalent to those of Qin et al. [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503] when the working propensity score model is correctly specified, and (b) doubly robust, like the augmented inverse probability weighting (AIPW) estimators of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Am Statist Assoc. 1994;89:846–866]. Thus, the proposed full likelihood-based estimators improve on the efficiency of the AIPW estimators when the working propensity score model is correct but the working conditional score model is possibly incorrect, and also improve on the empirical likelihood estimators of Qin, Zhang and Leung [Empirical likelihood in missing data problems. J Amer Statist Assoc. 2009;104:1492–1503] when the reverse is true, that is, the working conditional score model is correct but the working propensity score model is possibly incorrect. In addition, we consider a regression method for estimation of the regression coefficients when the working conditional score model is correctly specified; the asymptotic variance of the resulting estimator is no greater than the semiparametric variance bound characterized by the theory of Robins et al. [Estimation of regression coefficients when some regressors are not always observed. J Amer Statist Assoc. 1994;89:846–866]. Finally, we compare the finite-sample performance of various estimators in a simulation study.     

Estimation of Finite Population Mean in Stratified Random Sampling with Two Auxiliary Variables under Double Sampling Design

In this article, a chain ratio-product type exponential estimator is proposed for estimating finite population mean in stratified random sampling with two auxiliary variables under double sampling design. Theoretical and empirical results show that the proposed estimator is more efficient than the existing estimators, i.e., usual stratified random sample mean estimator, Chand (1975) chain ratio estimator, Choudhary and Singh (2012) estimator, chain ratio-product-type estimator, Sahoo et al. (1993) difference type estimator, and Kiregyera (1984) regression-type estimator. Two data sets are used to illustrate the performances of different estimators.     

Calibration Estimator of a Rare Sensitive Attribute with Poisson Distribution

In this paper we consider the calibration procedure for a rare sensitive attribute with Poisson distribution which suggested by Land et al. (2012) using auxiliary information associated with the variable of interest. In the calibration procedure, we can use auxiliary information such as socio-demographical variables for the respondents of rare sensitive attribute questions from an external source, and then this estimator can be improved with respect to the problems of non coverage or non response. From the efficiency comparison study, we show that the calibrated Poisson RR estimators are more efficient than that of Land et al. (2012), when the known population cell and marginal counts of auxiliary information are used for the calibration procedure.     

Statistical inference under imputation for proportional hazard model with missing covariates

Missing covariate data are common in biomedical studies. In this article, by using the non parametric kernel regression technique, a new imputation approach is developed for the Cox-proportional hazard regression model with missing covariates. This method achieves the same efficiency as the fully augmented weighted estimators (Qi et al. 2005. Journal of the American Statistical Association, 100:1250) and has a simpler form. The asymptotic properties of the proposed estimator are derived and analyzed. The comparisons between the proposed imputation method and several other existing methods are conducted via a number of simulation studies and a mouse leukemia data.     

A new procedure for variance estimation in simple random sampling using auxiliary information

In this article we have envisaged an efficient generalized class of estimators for finite population variance of the study variable in simple random sampling using information on an auxiliary variable. Asymptotic expressions of the bias and mean square error of the proposed class of estimators have been obtained. Asymptotic optimum estimator in the proposed class of estimators has been identified with its mean square error formula. We have shown that the proposed class of estimators is more efficient than the usual unbiased, difference, Das and Tripathi (Sankhya C 40:139–148, 1978), Isaki (J. Am. Stat. Assoc. 78:117–123, 1983), Singh et al. (Curr. Sci. 57:1331–1334, 1988), Upadhyaya and Singh (Vikram Math. J. 19:14–17, 1999b), Kadilar and Cingi (Appl. Math. Comput. 173:2, 1047–1059, 2006a) and other estimators/classes of estimators. In the support of the theoretically results we have given an empirical study.     

Consistency of The MMGLE under the piecewise proportional hazards models with interval-censored data

Wong et al. [(2018), ‘Piece-wise Proportional Hazards Models with Interval-censored Data’, Journal of Statistical Computation and Simulation, 88, 140–155] studied the piecewise proportional hazards (PWPH) model with interval-censored (IC) data under the distribution-free set-up. It is well known that the partial likelihood approach is not applicable for IC data, and Wong et al. (2018) showed that the standard generalised likelihood approach does not work either. They proposed the maximum modified generalised likelihood estimator (MMGLE) and the simulation results suggest that the MMGLE is consistent. We establish the consistency and asymptotically normality of the MMGLE.     

On the Mean Value Theorem for Estimating Functions

Abstract

Feng et al. revealed that the usual mean value theorem (MVT) should not be applied directly to a vector-valued function (e.g., the score function or a general estimating function under a multiparametric model). This note shows that the application of the Cramer–Wold’s device to a corrected version of the MVT is sufficient to obtain standard asymptotics for the estimators attained from vector-valued estimating functions.     

An improved cluster two-stage randomized response model

In this paper, Abdelfatah and Mazloum's (2015) two-stage randomized response model is extended to unequal probability sampling and stratified unequal probability sampling, both with and without replacement. The extended models result in more efficient estimators than Lee et al.'s (2014) estimators of the proportion of the population having a sensitive attribute.     

Statistical inference on asymptotic properties of two estimators for the partially linear single-index models

The outer product of gradients (OPG) estimation procedure based on least squares (LS) approach has been presented by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] to estimate the single-index parameter in partially linear single-index models (PLSIM). However, its asymptotic property has not been established yet and the efficiency of LS-based method can be significantly affected by outliers and heavy-tailed distributions. In this paper, we firstly derive the asymptotic property of OPG estimator developed by Xia et al. [An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B. 2002;64:363–410] in theory, and a novel robust estimation procedure combining the ideas of OPG and local rank (LR) inference is further developed for PLSIM along with its theoretical property. Then, we theoretically derive the asymptotic relative efficiency (ARE) of the proposed LR-based procedure with respect to LS-based method, which is shown to possess an expression that is closely related to that of the signed-rank Wilcoxon test in comparison with the t-test. Moreover, we demonstrate that the new proposed estimator has a great efficiency gain across a wide spectrum of non-normal error distributions and almost not lose any efficiency for the normal error. Even in the worst case scenarios, the ARE owns a lower bound equalling to 0.864 for estimating the single-index parameter and a lower bound being 0.8896 for estimating the nonparametric function respectively, versus the LS-based estimators. Finally, some Monte Carlo simulations and a real data analysis are conducted to illustrate the finite sample performance of the estimators.     

A note on nonparametric estimation of circular conditional densities

ABSTRACT

The conditional density offers the most informative summary of the relationship between explanatory and response variables. We need to estimate it in place of the simple conditional mean when its shape is not well-behaved. A motivation for estimating conditional densities, specific to the circular setting, lies in the fact that a natural alternative of it, like quantile regression, could be considered problematic because circular quantiles are not rotationally equivariant. We treat conditional density estimation as a local polynomial fitting problem as proposed by Fan et al. [Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika. 1996;83:189–206] in the Euclidean setting, and discuss a class of estimators in the cases when the conditioning variable is either circular or linear. Asymptotic properties for some members of the proposed class are derived. The effectiveness of the methods for finite sample sizes is illustrated by simulation experiments and an example using real data.     

Bayesian Approach in Nonparametric Count Regression with Binomial Kernel

Recently, Kokonendji et al. have adapted the well-known Nadaraya–Watson kernel estimator for estimating the count function m in the context of nonparametric discrete regression. The authors have also investigated the bandwidth selection using the cross-validation method. In this article, we propose a Bayesian approach in the context of nonparametric count regression for estimating the bandwidth and the variance of the model error, which has not been estimated in Kokonendji et al. The model error is considered as Gaussian with mean of zero and a variance of σ². The Bayes estimates cannot be obtained in closed form and then, we use the well-known Markov chain Monte Carlo (MCMC) technique to compute the Bayes estimates under the squared errors loss function. The performance of this proposed approach and the cross-validation method are compared through simulation and real count data.