Revisit of a randomized response model for estimating a rare sensitive attribute under probability proportional to size sampling using Poisson probability distribution

The present article deals with the estimation of mean number of individuals possess a rare sensitive attribute using Poisson probability distribution, when the population consists of clusters. Unbiased estimation procedures for the mean number of individuals have been suggested and their properties are discussed when the parameter of a rare non-sensitive unrelated attribute is assumed to be known as well as unknown. The suggested estimation procedure is further discussed for situation of stratified cluster population. Empirical studies are carried out to show the dominance of proposed method and resultant estimators over a well-known contemporary estimator.     

Robust-efficient fitting of mixed linear models: Methodology and theory

In many areas of application mixed linear models serve as a popular tool for analyzing highly complex data sets. For inference about fixed effects and variance components, likelihood-based methods such as (restricted) maximum likelihood estimators, (RE)ML, are commonly pursued. However, it is well-known that these fully efficient estimators are extremely sensitive to small deviations from hypothesized normality of random components as well as to other violations of distributional assumptions. In this article, we propose a new class of robust-efficient estimators for inference in mixed linear models. The new three-step estimation procedure provides truncated generalized least squares and variance components' estimators with hard-rejection weights adaptively computed from the data. More specifically, our data re-weighting mechanism first detects and removes within-subject outliers, then identifies and discards between-subject outliers, and finally it employs maximum likelihood procedures on the “clean” data. Theoretical efficiency and robustness properties of this approach are established.     

Estimation of the Moment Generating Function

A number of statistical problems use the moment generating function (mgf) for purposes other than determining the moments of a distribution. If the distribution is not completely specified, then the mgf must be estimated from available data. The empirical mgf makes no assumptions concerning the underlying distribution except for the existence of the mgf. In contrast to the nonparametric approach provided by the empirical mgf, alternative estimators can be formed based on an assumed parametric model. Comparison of these approaches is considered for two parametric models; the normal and a one parameter gamma. Comparison criteria are efficiency and empirical confidence interval coverage. In general the parametric estimators outperform the empirical mgf when the model is correct. The comparisons are extended to underlying models which are two component mixtures from the distributional family assumed by the parametric estimators. Under the mixture models the superiority of the parametric estimator depends upon the model, value of the argument of the mgf, and the comparison criterion. The empirical mgf is the better estimator in some cases.     

On the relevance of finite sample distribution theory

The implications of finite sample distribution theory for applied econometrics are explored. In general, its relevance is limited by three considerations: (i) sampling errors are of secondary importance in practice, (ii) exact and approximate results depend on the unknown structural parameters of the problem, and (iii) results more accurate than the limiting distribution require further distributional assumptions about some unobservable. Despite these difficulties, some useful information simultaneous equations estimators are sketched.     

Using pseudometrics in kernel density estimation

Common kernel density estimators (KDE) are generalised, which involve that assumptions on the kernel of the distribution can be given. Instead of using metrics as input to the kernels, the new estimators use parameterisable pseudometrics. In general, the volumes of the balls in pseudometric spaces are dependent on both the radius and the location of the centre. To enable constant smoothing, the volumes of the balls need to be calculated and analytical expressions are preferred for computational reasons. Two suitable parametric families of pseudometrics are identified. One of them has common KDE as special cases. In a few experiments, the proposed estimators show increased statistical power when proper assumptions are made. As a consequence, this paper describes an approach, where partial knowledge about the distribution can be used effectively. Furthermore, it is suggested that the new estimators are adequate for statistical learning algorithms such as regression and classification.     

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.     

Shrinkage and Penalty Estimators of a Poisson Regression Model

In this paper we propose Stein‐type shrinkage estimators for the parameter vector of a Poisson regression model when it is suspected that some of the parameters may be restricted to a subspace. We develop the properties of these estimators using the notion of asymptotic distributional risk. The shrinkage estimators are shown to have higher efficiency than the classical estimators for a wide class of models. Furthermore, we consider three different penalty estimators: the LASSO, adaptive LASSO, and SCAD estimators and compare their relative performance with that of the shrinkage estimators. Monte Carlo simulation studies reveal that the shrinkage strategy compares favorably to the use of penalty estimators, in terms of relative mean squared error, when the number of inactive predictors in the model is moderate to large. The shrinkage and penalty strategies are applied to two real data sets to illustrate the usefulness of the procedures in practice.     

Empirical bayes estimation of the mean in a multivariate normal distribution

We consider the problem of estimating the mean vector of a multivariate normal distribution under a variety of assumed structures among the parameters of the sampling and prior distributions. We adopt a pragmatic approach. We adopt distributional familites, assess hyperparmeters, and adopt patterned mean and coveariance structures when it is relatively simple to do so; alternatively, we use the sample data to estimate hyperparameters of prior distributions when assessment is a formidable task; such as the task of assessing parameters of multidimensional problems. James-Stein-like estimators are found to result. In some cases, we've been abl to show that the estimators proposed uniformly dominate the MLE's when measured with respect to quadratic loss functions.     

Randomized response multivariate designs for categorical data

Three approaches to multivariate estimation for categorical data using randomized response (RR) are described. In the first approach, practical only for 2×2 contingency tables, a multi-proportions design is used. In the second approach, a separate RR trial is used for each variate and it is noted that the multi­variate design matrix of conditional probabilities is given by the Kroneeker product of the univariate design matrices of each trial, provided that the trials are independent of each other in a certain sense. The third approach requires only a single randomization and thus may be viewed as the use of vector response. Finally, a special-purpose bivariate design is presented.     

A Comparison of Alternative Models for the Demand for Medical Care

We have tested alternative models of the demand for medical care using experimental data. The estimated response of demand to insurance plan is sensitive to the model used. We therefore use a split-sample analysis and find that a model that more closely approximates distributional assumptions and uses a nonparametric retransformation factor performs better in terms of mean squared forecast error. Simpler models are inferior either because they are not robust to outliers (e.g., ANOVA, ANOCOVA), or because they are inconsistent when strong distributional assumptions are violated (e.g., a two-parameter Box-Cox transformation).     

On simulation and properties of the stable law

The stable distribution, in its many parametrizations, is central to many stochastic processes. Many random variables that occur in the study of Lévy processes are related to it. Good progress has been made recently for simulating various quantities related to the stable law. In this note, we survey exact random variate generators for these distributions. Many distributional identities are also reviewed.     

On concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and its application in estimation

In this paper, we consider concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and develop its distribution theory. Using ranked set sample obtained from the above distribution, unbiased estimators of the parameters associated with the study variate involved in it are generated. The best linear unbiased estimators (BLUEs) based on observations in the ranked set sample of those parameters as well have been derived. The efficiencies of the BLUEs relative to the respective unbiased estimators generated also have been evaluated.     

Empirical likelihood estimation for linear regression models with AR(p) error terms with numerical examples

Linear regression models are useful statistical tools to analyze data sets in different fields. There are several methods to estimate the parameters of a linear regression model. These methods usually perform under normally distributed and uncorrelated errors. If error terms are correlated the Conditional Maximum Likelihood (CML) estimation method under normality assumption is often used to estimate the parameters of interest. The CML estimation method is required a distributional assumption on error terms. However, in practice, such distributional assumptions on error terms may not be plausible. In this paper, we propose to estimate the parameters of a linear regression model with autoregressive error term using Empirical Likelihood (EL) method, which is a distribution free estimation method. A small simulation study is provided to evaluate the performance of the proposed estimation method over the CML method. The results of the simulation study show that the proposed estimators based on EL method are remarkably better than the estimators obtained from CML method in terms of mean squared errors (MSE) and bias in almost all the simulation configurations. These findings are also confirmed by the results of the numerical and real data examples.     

A composite class of estimators using scrambled response mechanism for sensitive population mean in successive sampling

This paper considers the problem of estimation of population mean of a sensitive characteristics using non-sensitive auxiliary variable at current move in two move successive sampling. The proposed estimator is studied under five different scrambled response models. Various estimators have been elaborated to be the member of the proposed class of estimators. The properties of the proposed estimators have been analysed. Many estimators belonging to the proposed class have been explored under five scrambled response models. In order to identify the scrambled model effect, the proposed composite class of estimators is compared to the direct methods. Respondents privacy protection have also been elaborated under different models. Theoretical results are supplemented with numerical demonstrations using real data. Simulation has been carried out to show the applicability of proposed estimators and hence suitable recommendations are forwarded.     

Semiparametric estimation and inference for distributional and general treatment effects

Summary.  There is a large literature on methods of analysis for randomized trials with noncompliance which focuses on the effect of treatment on the average outcome. The paper considers evaluating the effect of treatment on the entire distribution and general functions of this effect. For distributional treatment effects, fully non-parametric and fully parametric approaches have been proposed. The fully non-parametric approach could be inefficient but the fully parametric approach is not robust to the violation of distribution assumptions. We develop a semiparametric instrumental variable method based on the empirical likelihood approach. Our method can be applied to general outcomes and general functions of outcome distributions and allows us to predict a subject's latent compliance class on the basis of an observed outcome value in observed assignment and treatment received groups. Asymptotic results for the estimators and likelihood ratio statistic are derived. A simulation study shows that our estimators of various treatment effects are substantially more efficient than the currently used fully non-parametric estimators. The method is illustrated by an analysis of data from a randomized trial of an encouragement intervention to improve adherence to prescribed depression treatments among depressed elderly patients in primary care practices.     

Estimating a sensitive proportion through randomized response procedures based on auxiliary information

Randomized response techniques are widely employed in surveys dealing with sensitive questions to ensure interviewee anonymity and reduce nonrespondents rates and biased responses. Since Warner’s (J Am Stat Assoc 60:63–69, 1965) pioneering work, many ingenious devices have been suggested to increase respondent’s privacy protection and to better estimate the proportion of people, π _A , bearing a sensitive attribute. In spite of the massive use of auxiliary information in the estimation of non-sensitive parameters, very few attempts have been made to improve randomization strategy performance when auxiliary variables are available. Moving from Zaizai’s (Model Assist Stat Appl 1:125–130, 2006) recent work, in this paper we provide a class of estimators for π _A , for a generic randomization scheme, when the mean of a supplementary non-sensitive variable is known. The minimum attainable variance bound of the class is obtained and the best estimator is also identified. We prove that the best estimator acts as a regression-type estimator which is at least as efficient as the corresponding estimator evaluated without allowing for the auxiliary variable. The general results are then applied to Warner and Simmons’ model.     

Sampling properties of estimators of the log-logistic distribution with application to Canadian precipitation data

We consider the probability-weighted moment and the maximum-likelihood estimators of two parameters in the log-logistic distribution. Quantile estimators are obtained using both methods. The distributional properties of these estimators are studied in large samples, via asymptotic theory, and in small and moderate samples, via Monte Carlo simulation. The distribution is shown to be appropriate for a wide variety of meteorological data.     

Double Sampling Ratio-product Estimator of a Finite Population Mean in Sample Surveys

It is well known that two-phase (or double) sampling is of significant use in practice when the population parameter(s) (say, population mean X¯) of the auxiliary variate x is not known. Keeping this in view, we have suggested a class of ratio-product estimators in two-phase sampling with its properties. The asymptotically optimum estimators (AOEs) in the class are identified in two different cases with their variances. Conditions for the proposed estimator to be more efficient than the two-phase sampling ratio, product and mean per unit estimator are investigated. Comparison with single phase sampling is also discussed. An empirical study is carried out to demonstrate the efficiency of the suggested estimator over conventional estimators.     

A simple approach for generating correlated binary variates ∗

Correlated binary data arise frequently in medical as well as other scientific disciplines; and statistical methods, such as generalized estimating equation (GEE), have been widely used for their analysis. The need for simulating correlated binary variates arises for evaluating small sample properties of the GEE estimators when modeling such data. Also, one might generate such data to simulate and study biological phenomena such as tooth decay or periodontal disease. This article introduces a simple method for generating pairs of correlated binary data. A simple algorithm is also provided for generating an arbitrary dimensional random vector of non-negatively correlated binary variates. The method relies on the idea that correlations among the random variables arise as a result of their sharing some common components that induce such correlations. It then uses some properties of the binary variates to represent each variate in terms of these common components in addition to its own elements. Unlike most previous approaches that require solving nonlinear equations or use some distributional properties of other random variables, this method uses only some properties of the binary variate. As no intermediate random variables are required for generating the binary variates, the proposed method is shown to be faster than the other methods. To verify this claim, we compare the computational efficiency of the proposed method with those of other procedures.