A true simulation study of three estimators at equal protection of respondents in randomized response sampling

In this study, three different estimators for estimating the proportion of a sensitive attribute in survey sampling are compared at equal protection of the respondents. The three estimators considered are due to Odumade and Singh (2009, Comm. Statist. Theory Methods) , Singh and Sedory (2011, Sociological Methods and Research) and a new estimator obtained by minimizing a chi‐squared distance. A SAS Macro is developed to compare these three estimators using a simulation study at equal protection of the respondents. A set of data from a real face‐to‐face interview was collected using two decks of cards and has been analyzed. The results are discussed.     

Survey statisticians celebrate golden jubilee year 2003 of the linear regression estimator

In the present investigation, a general set-up for inference from survey data that covers the estimation of variance of estimators of totals and distribution functions has been considered, using known first and second order moments of auxiliary information at the estimation stage. The traditional linear regression estimator of population total owed to Hansen et al. Sample survey methods and theory. vol. 1 & 2, New York, Wiley (1953) is shown to be unique in its class of estimators, and celebrates Golden Jubilee Year-2003 for its outstanding performance in the literature by following Singh Advanced sampling theory with applications: How Michael selected Amy, vols 1 & 2, Kluwer, The Netherlands, pp 1–1247 2003. This particular paper has been designed to repair the methodology of Rao J. Off Stat 10(2):153–165 (1994) and hence that of Singh Ann Ins Stat Math 53(2):404–417 (2001). Although there is no need of simulation study to demonstrate the superiority of the proposed technique, because the theoretical results are crystal clear, but a small scale level simulation study have been designed to show the performance of the proposed estimators over the existing estimators in the literature.     

An optional scrambled randomized response technique for practical surveys

Eichhorn and Hayre (J Stat Plan Inference 7:307–316, 1983) introduced the scrambled response technique to gather information on sensitive quantitative variables. Singh and Joarder (Metron 15:151–157, 1997), Gupta et al. (J Stat Plan Inference 100:239–247, 2002) and Bar-Lev et al. (Metrika 60:255–260, 2004) permitted the respondents either to report their true values on the sensitive quantitative variable or the scrambled response and developed the optional randomized response (ORR) technique based on simple random sample with replacement (SRSWR). While developing the ORR procedure, these authors made the assumption that the probability of disclosing the true response or the randomized response (RR) is the same for all the individuals in a population. This is not a very realistic assumption as in practical survey situations the probability of reporting the true value or the RR generally varies from unit to unit. Moreover, if one generalizes the ORR method as developed by these authors relaxing the ‘constant probability’ assumption, the variance of an unbiased estimator for the population total or mean can not be estimated as this involves the unknown parameter, ‘the probability of revealing the true response’. Here we propose a modified ORR procedure for stratified unequal probability sampling after relaxing the assumption of ‘constant probability’ of providing the true response. It is also demonstrated with a numerical exercise that our procedure produces better estimator for a population total than that provided by the method suggested by the earlier authors.     

Properties of h-Likelihood Estimators in Clustered Data

We study properties of the maximum h-likelihood estimators for random effects in clustered data. To define optimality in random effects predictions, several foundational concepts of statistics such as likelihood, unbiasedness, consistency, confidence distribution and the Cramer–Rao lower bound are extended. Exact probability statements about interval estimators for random effects can be made asymptotically without a prior assumption. Using the binary-matched pair example, we illustrated that the use of random effects recover information, leading to the boon on estimating treatment effects.     

Estimation of population proportion in randomized response sampling using weighted confidence interval construction

This paper considers the problem of estimation for binomial proportions of sensitive attributes in the population of interest. Randomized response techniques are suggested for protecting the privacy of respondents and reducing the response bias while eliciting information on sensitive attributes. By applying the Wilson (J Am Stat Assoc 22:209–212, 1927) score approach for constructing confidence intervals, various probable point estimators and confidence interval estimators are suggested for the common structures of randomized response procedures. In addition, efficiency comparisons are carried out to study the performances of the proposed estimators for both the cases of direct response surveys and randomized response surveys. Circumstances under which each proposed estimators is better are also identified.     

Estimation from two-stage unequal probability sampling with missing units

A formula is presented for an unbiased estimator for the variance of an unbiased estimator of a survey population total as well as for an unbiased estimator of its variance based on sampling in two-stages following Rao et al. J Roy Stat Soc B 24: 482–491 (1962) scheme in both stages when the originally selected units in both stages cannot be fully covered in the survey but are to be randomly sub-sampled. The development is helpful to tackle non-responses if assumed to have occurred at random in either or both the stages     

Estimation for sensitive characteristics using optional randomized response technique

This paper considers the problem of procuring honest responses to estimate the population mean of sensitive quantitative characteristics. It aims at pointing out that potential difference between sub-populations seems to be unnoticed under the usual optional randomized response technique. To make an extension to cover all the situations, we propose two alternative survey procedures that enable us to estimate the mean and the variance unbiasedly. Estimation for the sensitivity level of a question is under consideration as well. Efficiency comparisons have also been carried out to study the performances of the proposed procedures.     

Detecting Untruthful Answering in Randomized Response Sampling

This paper considers the problem of unbiasedly estimating the population proportion when the study variable is potential sensitive in nature. In order to protect the respondent’s privacy, various techniques of generating randomized response rather than direct response are available in the literature. But the theory concerning them is developed only under the hypothesis of completely truthful reporting. Actually, the occurrence of untruthful reporting is a prospect in dealing with highly sensitive matters such as abortion or socially deviant behaviors. Illustrating Warner’s [(1965), Journal of the American Statistical Association. 60: 63–69] randomized response technique we show how unbiased estimation of the population proportion can be extended to cover a case when some respondents may lie.     

Distribution of personal income: Development of a new model and its application to U.S. income data

This paper proposes a four-parameter statistical model of the personalized distribution of income using the ‘income share elasticity’ approach suggested by Esteban (1986). Our proposed model includes the Singh–Maddala (1976) and Dagum (1977) distributions as special cases. The generalized beta II distribution of McDonald (1984) is also a variant of this model. It appears to give an excellent fit to US income data and its empirical performance turns out to be superior to those of the Singh–Maddala (1976), Dagum (1977), the five-parameter Champernowne (1953) distributions and the generalized beta II distribution of McDonald (1984) on some data.     

Imputation methods of missing data for estimating the population mean using simple random sampling with known correlation coefficient

This paper considers three ratio estimators of the population mean using known correlation coefficient between the study and auxiliary variables in simple random sample when some sample observations are missing. The suggested estimators are compared with the estimators of Singh and Horn (Metrika 51:267–276, 2000), Singh and Deo (Stat Pap 44:555–579, 2003) and Kadilar and Cingi (Commun Stat Theory Methods 37:2226–2236, 2008). They are compared with other imputation estimators based on the mean or a ratio. It is found that the suggested estimators are approximately unbiased for the population mean. Also, it turns out that the suggested estimators perform well when compared with the other estimators considered in this study.     

Density estimation for nonlinear parametric models with conditional heteroscedasticity

This article studies density and parameter estimation problems for nonlinear parametric models with conditional heteroscedasticity. We propose a simple density estimate that is particularly useful for studying the stationary density of nonlinear time series models. Under a general dependence structure, we establish the root n$n$ consistency of the proposed density estimate. For parameter estimation, a Bahadur type representation is obtained for the conditional maximum likelihood estimate. The parameter estimate is shown to be asymptotically efficient in the sense that its limiting variance attains the Cramér–Rao lower bound. The performance of our density estimate is studied by simulations.     

Mean square error estimation in multi-stage sampling

Summary: Suppose for a homogeneous linear unbiased function of the sampled first stage unit (fsu)-values taken as an estimator of a survey population total, the sampling variance is expressed as a homogeneous quadratic function of the fsu-values. When the fsu-values are not ascertainable but unbiased estimators for them are separately available through sampling in later stages and substituted into the estimator, Raj (1968) gave a simple variance estimator formula for this multi-stage estimator of the population total. He requires that the variances of the estimated fsu-values in sampling at later stages and their unbiased estimators are available in certain `simple forms'. For the same set-up Rao (1975) derived an alternative variance estimator when the later stage sampling variances have more ‘complex forms’. Here we pursue with Raj's (1968) simple forms to derive a few alternative variance and mean square error estimators when the condition of homogeneity or unbiasedness in the original estimator of the total is relaxed and the variance of the original estimator is not expressed as a quadratic form.  We illustrate a particular three-stage sampling strategy and present a simulation-based numerical exercise showing the relative efficacies of two alternative variance estimators. Received: 19 February 1999     

A two-stage stratified Warner’s randomized response model using optimal allocation

This paper applies Kim and Wardes (2004) stratified Warners randomized response model to Mangat and Singhs (1990) two-stage randomized response model. The proposed stratified randomized response model has an optimal allocation and a large gain in precision. Hence, the estimator based on the proposed method is more efficient than Kim and Wardes (2004) and Mangat and Singhs (1990) estimators under the conditions presented in both the case of completely truthful reporting and that of not completely truthful reporting by the respondents.     

A generalized randomized response technique

To eliminate a major source of bias in surveys of human populations resulting from respondents refusal to cooparate in cases where a question of sensitive nature is involved, the idea of “randomized response” was introduced by Warner (1965). In this paper, an alternative randomized response technique is presented which improves upon the pioneering work of Warner (1965). The procedure includes Warner's method as a special case for a specific choice of the parameters. In addition, a generalization of the proposed method is presented.     

Estimating regression coefficients from survey data by asymptotic design-cum-model based approach

Postulating a linear regression of a variable of interest on an auxiliary variable with values of the latter known for all units of a survey population, we consider appropriate ways of choosing a sample and estimating the regression parameters. Recalling Thomsen’s (1978) results on non-existence of ‘design-cum-model’ based minimum variance unbiased estimators of regression coefficients we apply Brewer’s (1979) ‘asymptotic’ analysis to derive ‘asymptotic-design-cummodel’ based optimal estimators assuming large population and sample sizes. A variance estimation procedure is also proposed.     

On two properties of an unequal probability sampling scheme

Summary For an inclusion probability proportional to size (IPPS) sampling scheme recently proposed by Saxena, Singh and Srivastava (1986), it is shown that under certain simple verifiable conditions (1) the Horvitz-Thompson (1952) estimator based on it has a smaller variance than the variance of the Hansen-Hurwitz (1943) estimator based on probability proportional to size (PPS) sampling with replacement (WR) both involving the same size-measures and the expected sample size in the former being equal to the number of draws in the latter and (2) the Yates-Grundy (1953) estimator for the variance of the Horvitz-Thompson estimator based on this IPPS scheme is uniformly non-negative.     

EWMA-GARCH模型与GARCH模型在估计收益率波动上的差异的实证及理论分析

VaR作为衡量风险的指标,其核心则在于对波动,亦即方差的估计。基于时间序列,关于条件方差的经典模型是GARCH模型,尽管后来又衍生出了EGARCH,PARCH等复杂模型,但在实务中GARCH模型仍占有重要的地位。文章分析了一种比较新的结合了EWMA模型的GARCH模型(以下称为EWMA-GARCH模型)计算VaR的参数估计方法,以检验其在估计波动上的实用性,并对实证检验结果做了理论分析。分析结果表明,尽管该结合模型缺乏完整的理论支持,但是其计算效果仍比较良好,当然这样良好的结果是建立在因缺乏理论依据而导致的对模型的其他要求之上的.至于是采用受理论支持的模型还是并不输实践价值的模型,文章也给出了一定的建议。     

Design-based distribution function estimation for stigmatized populations

In this paper, we discuss in a general framework the design-based estimation of population parameters when sensitive data are collected by randomized response techniques. We show in close detail the procedure for estimating the distribution function of a sensitive quantitative variable and how to estimate simultaneously the population prevalence of individuals bearing a stigmatizing attribute and the distribution function for the members belonging to the hidden group. The randomized response devices by Greenberg et al. (J Am Stat Assoc 66:243–250, 1971), Franklin (Commun Stat Theory Methods 18:489–505, 1989), and Singh et al. (Aust NZ J Stat 40:291–297 1998) are here considered as data-gathering tools.     

On the optimum number of levels in the one-way model with random effects

The optimal number of levels is studied for the one-way random model with normally distributed effects. The optimum criteria used are based on the variances of the traditional analysis of variance estimators of the variance components. Exact solutions are compared to earlier results based on lower bounds of the sampling variances. Comparisons are also made to the large-sample variances of the estimates based on restricted maximum likelihood. Received February 2002