Pseudo-Bayes and pseudo-empirical Bayes estimators in randomized response sampling

In this article, new pseudo-Bayes and pseudo-empirical Bayes estimators for estimating the proportion of a potentially sensitive attribute in a survey sampling have been introduced. The proposed estimators are compared with the recent estimator proposed by Odumade and Singh [Efficient use of two decks of cards in randomized response sampling, Comm. Statist. Theory Methods 38 (2009), pp. 439–446] and Warner [Randomized response: A survey technique for eliminating evasive answer bias, J. Amer. Statist. Assoc. 60 (1965), pp. 63–69].     

On using non identical and independent Bernoulli trials in human surveys

Unlike the usual randomized response techniques, as a pioneering attempt, this article focuses on using non identical independent Bernoulli trials in sensitive surveys. For this purpose, a general class of randomized response techniques is considered. The usual randomized response techniques are based on a fixed probability of having a yes answer. Contrary to usual techniques, in the proposed technique every respondent has a different probability of reporting a yes answer. With this setting, in most of the situations, the proposed technique is observed performing better in terms of variability. To illustrate and support the superiority of the proposed technique it is compared with models such as Warner (1965), Greenberg et al. (1969), Mangat and Singh (1990), and Mangat (1994) using identical Bernoulli trials. Relative efficiency and privacy protection are studied in detail using Warner (1965) and Mangat (1994) models.     

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.     

Use of Two Decks of Cards in Randomized Response Techniques for Complex Survey Designs

The randomized response (RR) technique with two decks of cards proposed by Odumade and Singh (2009 Odumade , O. , Singh , S. ( 2009 ). Efficient use of two deck of cards in randomized response sampling . Commun. Statist. Theor. Meth. 38 : 439 – 446 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) can always be made more efficient than the RR techniques proposed by Warner (1965 Warner , S. L. ( 1965 ). Randomize response: A survey technique for eliminating evasive answer bias . J. Amer. Statist. Assoc. 60 : 63 – 69 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Mangat and Singh (1990 Mangat , N. S. , Singh , R. ( 1990 ). An alternative randomized response procedure . Biometrika 77 : 349 – 442 .[Crossref], [Web of Science ®] , [Google Scholar]), and Mangat (1994 Mangat , N. S. ( 1994 ). An improved randomized response strategy . J. Roy. Statist. Soc. B 56 : 93 – 95 . [Google Scholar]) by adjusting the proportion of cards in the decks. The proposed method of Odumade and Singh (2009 Odumade , O. , Singh , S. ( 2009 ). Efficient use of two deck of cards in randomized response sampling . Commun. Statist. Theor. Meth. 38 : 439 – 446 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is limited to simple random sampling with replacement (SRSWR) sampling only. In this article, generalization of Odumade and Singh strategy is provided for complex survey designs and a wider class of estimators. The results of Odumade and Singh (2009 Odumade , O. , Singh , S. ( 2009 ). Efficient use of two deck of cards in randomized response sampling . Commun. Statist. Theor. Meth. 38 : 439 – 446 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) can be derived from the proposed method as a special case.     

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).     

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.     

Generalized geometric distribution of order k: A flexible choice to randomize the response

This article focuses on the improvement of a well-celebrated randomized response technique of Kuk. A generalized randomized response technique is suggested. In particular, the generalized geometric distribution of order k is introduced as a randomization device for estimating the population proportion of a rare sensitive attribute. The proposed randomized response technique includes Singh and Grewal and Hussain et al. techniques as its special cases. Through numerical illustrations, it is established that the suggested technique is superior to the Kuk, Singh and Grewal, and Hussain et al. techniques. Flexibility of the proposed technique is also discussed.     

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.     

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.     

Comments on the paper “Bias-adjustment and calibration of jackknife variance estimator in the presence of non-response”

Singh and Arnab (2010) presented a bias adjustment to the jackknife variance estimator of Rao and Sitter (1995) in the presence of non-response. In their paper, they obtained a second-order approximation of the bias of the Rao-Sitter variance estimator and then proposed a bias-adjusted estimator based on this approximation. To compare their proposed variance estimator to various other variance estimators, they performed a simulation study and showed that their variance estimator is superior to the Rao-Sitter variance estimator. In fact they showed that the Rao-Sitter variance estimator suffers from severe underestimation. These results contradict those in the literature, which indicate that the Rao-Sitter variance estimator suffers from a positive bias if the sampling fractions are not negligible; see Rao and Sitter (1995), Lee et al. (1995) and Haziza and Picard (2011). Because of this contradiction, we felt that a further investigation was warranted. In this paper, we attempt to recreate the results of Singh and Arnab (2010) and, in fact, show that their second order approximation to the bias of the Rao-Sitter variance estimator is incorrect and that their simulation results are also questionable.     

A two-stage unrelated randomized response model for estimating a rare sensitive attribute in probability proportional to size sampling using Poisson distribution

This paper describes the estimating procedures of mean number of entities that possess a rare sensitive attribute using the Mangat (1992) randomized device, when the population consists of some clusters and the population is again stratified with some clusters in each stratum. Unbiased estimation procedures for the mean number of individuals have been discussed and their properties are described when the parameter of a rare unrelated attribute is assumed to be known and unknown. An empirical study is carried out to show the dominance of the proposed estimator over Lee et al. (2013) estimator.     

A new estimator of proportion with a linear function using data from two-decks randomized response model

A new estimator for estimating the proportion of a potentially sensitive attribute in survey sampling has been introduced by solving a linear equation. The proposed estimator has been compared with the estimator proposed by Odumade and Singh (2009) with equal protection to all of the respondents. The asymptotic properties of the proposed estimator are investigated through exact numerical illustrations for different choices of parameters. A non randomized response approach has been suggested. A scope for further research has also been pointed out.     

A GENERALIZED TWO-STAGE RANDOMIZED RESPONSE PROCEDURE IN COMPLEX SAMPLE SURVEYS

The randomized response (RR) technique pioneered by Warner, S.L. (1965) [Randomised response: a survey technique for eliminating evasive answer bias. J. Amer. Statist. Assoc. 60 , 63–69] is a useful tool in estimating the proportion of persons in a community bearing sensitive or socially disapproved characteristics. Mangat, N.S. & Singh, R. (1990) [An alternative rendomized response procedure. Biometrika 77 , 439–442] proposed a modification of Warner's procedure by using two RR techniques. Presented here is a generalized two‐stage RR procedure and derivation of the condition under which the proposed procedure produces a more precise estimator of the population parameter. A comparative study on the performance of this two‐stage procedure and conventional RR techniques, assuming that the respondents' jeopardy level in this proposed method remains the same as that offered by the traditional RR procedures, is also reported. In addition, a numerical example compares the efficiency of the proposed method with the traditional RR procedures.     

A stratified Warner''s randomized response model

This paper proposes a new stratified randomized response model based on Warner's (J. Amer. Statist. Assoc. 60 (1965) 63) model that has an optimal allocation and large gain in precision. It also presents a drawback of the Hong et al. (Korean J. Appl. Statist. 7 (1994) 141) model under their proportional sampling assumption. It is shown that the proposed model is more efficient than the Hong et al. (Korean J. Appl. Statist. 7 (1994) 141) stratified randomized response model. Additionally, it is shown that the estimator based on the proposed method is more efficient than the Warner (J. Amer. Statist. Assoc. 60 (1965) 63), the Mangat and Singh (Biometrika 77 (1990) 439) and the Mangat (J. Roy. Statist. SQC. Ser. B 56 (1) (1994) 93) estimators under the conditions presented in both the case of completely truthful reporting and that of not completely truthful reporting by the respondents.     

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 mixed randomized response model

To implement the privacy problem with the Moors (J. Amer. Statist. Assoc. 66 (1971) 627) model, Mangat et al. (Commun. Statist. Theory Methods 26(3) (1997) 243) and Singh et al. (J. Statist. Plann. Inference 83 (2000) 243) presented several strategies as an alternative to Moors model, but their models may lose a large portion of data information and require a high cost to obtain confidentiality of the respondents. Our proposed model has the advantage of simplicity over the previous models while protecting confidentiality. We extend the proposed model to stratified sampling.     

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.     

An improved two stage randomized response strategy

Mangat and Singh (1990) have suggested a two stage randomized response technique to estimate the proportion of population possessing a sensitive attribute. The procedure was shown to be more efficient than the procedure due to Warner (1965). Recently, Tracy and Osahan (1993) have suggested a modification to the Mangat and Singh (1990) procedure which results in a more efficient strategy in practice. In this paper we propose a modification to the Tracy and Osahan (1993) procedure. The modified procedure is a generalization of Tracy and Osahan (1993) and is always more efficient than their strategy. An empirical study has also been undertaken to find the extent of relative efficiency.     

Mean and sensitivity estimation of a sensitive variable through additive scrambling

Abstract

This article focuses on reducing the additional variance due to randomization of the responses. The idea of additive scrambling and its inverse has been used along with (i) split sample approach and (ii) double response approach. Specifically, our proposal is based on Gupta et al. (2006) randomized response model. We selected this model for improvement because it provides estimator of mean and sensitivity level of a sensitive variable and is better than all of its competitors proposed earlier to it and even Gupta et al. (2006) sensitivity estimator is better than that of Gupta et al. (2010). Our suggested estimators are unbiased estimators and perform better than Gupta et al. (2006) estimator. The issue of privacy protection is also discussed.     

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.