A stratified unrelated question randomized response model

This paper presents a new randomized response model that combines Kim and Warde's (2004) stratified Warner's randomized response technique using optimal allocation with the unrelated question randomized response model. The empirical studies performed show that, for the prior information given, the new model is more efficient in terms of variance (in the case of completely truthful reporting) and mean square error (in the case of less than completely truthful reporting) than its component models.     

A stratified unrelated question randomized response model using Neyman allocation

This paper applies stratified random sampling using Neyman allocation to Mangat et al. (1992 Mangat, N.S., Singh, R., Singh, S. (1992). An improved unrelated question randomized response strategy. Cal. Stat. Assoc. Bull. 42:277–281.[Crossref] , [Google Scholar]) unrelated question randomized response (RR) strategy for both completely truthful reporting and less than completely truthful reporting. It is shown that, for the prior information given, our new model is more efficient in terms of variance (in the case of completely truthful reporting) and mean square error (in terms of less than completely truthful reporting) than Kim and Elam's (2007 Kim, J.M., Elam, M.E. (2007). A stratified unrelated question randomized response model. Stat. Papers 48:215–233.[Crossref], [Web of Science ®] , [Google Scholar]) model. Numerical illustrations and graphs are also given in support of the present study.     

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.     

A TWO-STAGE UNRELATED RANDOMISED RESPONSE PROCEDURE

In a sample survey, questions requiring personal or controversial assertions often give rise to resistance. A randomised response procedure can be used to help the researcher gather accurate data in this case. This paper describes a new two-stage unrelated randomised response procedure that combines the use of two randomisation devices (Mangat & Singh, 1990) and an unrelated question (Horwitz et al. 1967). It examines the situation where the respondents are not completely truthful in their answers. The efficiency of this new method is compared with the original one-stage procedure proposed by Horwitz et al. (1967), and guidelines for choosing the values of different parameters for the procedures are provided. Results from an empirical study which examines the efficiency and feasibility of the proposed method are given.     

Standby redundancies at component level versus system level in series system

Abstract

It is known that the redundancy at the component level is better than the system level for the case of active redundancy. However, few results are available for standby redundancy due to the complexity of convolution. This note stochastically compares allocations of standby redundancies in series systems with exponential components at the component level versus the system level in sense of the likelihood ratio ordering. The established results strengthen and extend some known ones in the literature.     

The Extremal Dependence Measure and Asymptotic Independence

Abstract

Extremal dependence analysis assesses the tendency of large values of components of a random vector to occur simultaneously. This kind of dependence information can be qualitatively different than what is given by correlation which averages over the total body of the joint distribution. Also, correlation may be completely inappropriate for heavy tailed data. We study the extremal dependence measure (EDM), a measure of the tendency of large values of components of a random vector to occur simultaneously and show consistency of an estimator of the EDM. We also show asymptotic normality of an idealized estimator in a restricted case of multivariate regular variation where scaling functions do not have to be estimated.     

Estimation of a population mean with two-way stratification using a systematic allocation scheme

A technique of systematically allocating a sample to the strata formed by double stratification is presented. The method can proportionally allocate the sample along each variable of stratification. If there are R strata and C strata for the first and second variable of stratification respectively, the technique requires that the total sample size be at least as large as max(R, C). An unbiased estimator of the population mean is given and its variance is obtained. The technique is compared with a random allocation procedure given by Bryant, Hartley, and Jessen (1960). Numerical examples are given suggesting when one technique is superior to the other.     

Some New Results on the Multinomial Randomized Response Model

ABSTRACT

The randomized response technique is an effective survey method designed to elicit sensitive information while ensuring the privacy of the respondents. In this article, we present some new results on the randomization response model in situations wherein one or two response variables are assumed to follow a multinomial distribution. For a single sensitive question, we use the well-known Hopkins randomization device to derive estimates, both under the assumption of truthful and untruthful responses, and present a technique for making pairwise comparisons. When there are two sensitive questions of interest, we derive a Pearson product moment correlation estimator based on the multinomial model assumption. This estimator may be used to quantify the linear relationship between two variables when multinomial response data are observed according to a randomized-response protocol.     

A stratified Tracy and Osahan's two-stage randomized response model

ABSTRACT

This paper proposes an alternative two-stage stratified randomized response model based on Tracy and Osahan (1999 Tracy, D.S., Osahan, S.S. (1999). An improved randomized response technique. Pak. J. Stat. 15(1):1–6. [Google Scholar]) model that has an optimal allocation and large gain in precision. It is also shown that the proposed model is more efficient than Kim and Warde (2004 Kim, J., Warde, W. (2004). A stratified Warner randomized response model. J. Stat. Plan. Infer. 120:155–165.[Crossref], [Web of Science ®] , [Google Scholar]) and Kim and Elam (2005 Kim, J.M., Elam, M.E. (2005). A two-stage stratified Warner's randomized response model using optimal allocation. Metrika 61:1–7.[Crossref], [Web of Science ®] , [Google Scholar]) stratified randomized response models under the conditions presented in both the cases of completely truthful reporting and that of not completely truthful reporting by the respondents. Numerical illustrations and graphs are also given in support of the present study.     

A note on the estimates of multivariate Gaussian probability

ABSTRACT

Some lower and upper bounds of multivariate Gaussian probability are given based on the univariate Mills’ ratio. These bounds are sharper than known ones on the multivariate Mills’ ratio in many case.     

TWO-DIMENSIONAL BALANCED SAMPLING PLANS EXCLUDING CONTIGUOUS UNITS

ABSTRACT

A balanced sampling plan excluding contiguous units (or BSEC for short) was first introduced by Hedayat, Rao and Stufken in 1988. These designs can be used for survey sampling when the units are arranged in one-dimensional ordering and the contiguous units in this ordering provide similar information. In this paper, we generalize the concept of a BSEC to the two-dimensional situation and give constructions of two-dimensional BSECs with block size 3. The existence problem is completely solved in the case where λ = 1.     

On pre-test estimation of parametric functions in the general Gauss-Markov model with quadratic risk

The approach to preliminary test estimation based on comparing the weighted quadratic risk function of two competing estimators of β under the linear regression model {y,Xβ, σ² I} is extended to the case when a given vector of parametric functions κ=Kβ is to be estimated under the general Gauss-Markov model.     

On the Expressions of Estimability in Testing General Linear Hypotheses

In testing a general linear hypothesis of the form K ′β ? ( W ′) under a general linear model, an equivalent hypothesis involving only estimable parametric functions is provided, and then an explicit test statistic in terms of the model matrices is given. The corresponding results are expanded to the case of a general linear model with a restriction and are illustrated by an example.     

Study of incompatibility or near compatibility of bivariate discrete conditional probability distributions through divergence measures

Consider a two-dimensional discrete random variable (X, Y) with possible values 1, 2, …, I for X and 1, 2, …, J for Y. For specifying the distribution of (X, Y), suppose both conditional distributions, of X given Y and of Y given X, are provided. Under this setting, we present here different ways of measuring discrepancy between incompatible conditional distributions in the finite discrete case. In the process, we also suggest different ways of defining the most nearly compatible distributions in incompatible cases. Many new divergence measures are discussed along with those that are already known for determining the most nearly compatible joint distribution P. Finally, a comparative study is carried out between all these divergence measures as some examples.     

The Effects of E-journal Management Tools and Services on Serials Cataloging

Abstract

E-journal management tools and services such as MARC record services, A-to-Z lists, and link resolvers are changing e-journal cataloging. This column explores these changes in the academic environment through interviews with ten librarians representing eight universities. Three areas of change in serials cataloging are explored: (1) changes to the MARC record, including how libraries are adding/creating MARC records for their catalogs, the number and type of MARC records being created and linking within MARC bibliographic and holdings records; (2) the manner in which serials catalogers are being informed of changes; and (3) the evolving role of the serials cataloger. Future trends and advice for evolving workflow practices conclude the discussion.     

Model fitting and inference under latent equilibrium processes

This paper presents a methodology for model fitting and inference in the context of Bayesian models of the type f(Y | X,θ)f(X|θ)f(θ), where Y is the (set of) observed data, θ is a set of model parameters and X is an unobserved (latent) stationary stochastic process induced by the first order transition model f(X ^(t+1)|X ^(t),θ), where X ^(t) denotes the state of the process at time (or generation) t. The crucial feature of the above type of model is that, given θ, the transition model f(X ^(t+1)|X ^(t),θ) is known but the distribution of the stochastic process in equilibrium, that is f(X|θ), is, except in very special cases, intractable, hence unknown. A further point to note is that the data Y has been assumed to be observed when the underlying process is in equilibrium. In other words, the data is not collected dynamically over time. We refer to such specification as a latent equilibrium process (LEP) model. It is motivated by problems in population genetics (though other applications are discussed), where it is of interest to learn about parameters such as mutation and migration rates and population sizes, given a sample of allele frequencies at one or more loci. In such problems it is natural to assume that the distribution of the observed allele frequencies depends on the true (unobserved) population allele frequencies, whereas the distribution of the true allele frequencies is only indirectly specified through a transition model. As a hierarchical specification, it is natural to fit the LEP within a Bayesian framework. Fitting such models is usually done via Markov chain Monte Carlo (MCMC). However, we demonstrate that, in the case of LEP models, implementation of MCMC is far from straightforward. The main contribution of this paper is to provide a methodology to implement MCMC for LEP models. We demonstrate our approach in population genetics problems with both simulated and real data sets. The resultant model fitting is computationally intensive and thus, we also discuss parallel implementation of the procedure in special cases.     

Best quadratic predictions in finite populations

For a finite population and the resulting linear model Y=Xβ+e, the problem of the optimal invariant quadratic predictors including optimal invariant quadratic unbiased predictor and optimal invariant quadratic (potentially) biased predictor for the population quadratic quantities, f(H)=Y′HY , is of interest and has been previously considered in the literature for the case of HX=0. However, the special case does not contain all of situations at all. So, predicting f(H) in general situations may be of particular interest. In this paper, we make an effort to investigate how to offer a good predictor for f(H), not restricted yet to the mentioned case. Permutation matrix techniques play an important role in handling the process. The expected predictors are finally derived. In addition, we mention that the resulting predictors can be viewed as acceptable in all situations.     

Bayes and Empirical Bayes Inference in Changepoint Problems

ABSTRACT

In the case of the random design nonparametric regression, the double smoothing technique is applied to estimate the multivariate regression function. The proposed estimator has desirable properties in both the finite sample and the asymptotic cases. In the finite sample case, it has bounded conditional (and unconditional) bias and variance. On the other hand, in the asymptotic case, it has the same mean square error as the local linear estimator in Fan (Design-Adaptive Nonparametric Regression. Journal of the American Statistical Association 1992, 87, 998–1004; Local Linear Regression Smoothers and Their Minimax Efficiencies. Annals of Statistics 1993, 21, 196–216). Simulation studies demonstrate that the proposed estimator is better than the local linear estimator, because it has a smaller sample mean integrated square error and gives smoother estimates.     

Bounds for L-Statistics from Weakly Dependent Samples of Random Length

ABSTRACT

Sharp bounds on expected values of L-statistics based on a sample of possibly dependent, identically distributed random variables are given in the case when the sample size is a random variable with values in the set {0, 1, 2,…}. The dependence among observations is modeled by copulas and mixing. The bounds are attainable and provide characterizations of some non trivial distributions.     

A very simple proof of the multivariate Chebyshev's inequality

Abstract

In this short note, a very simple proof of the Chebyshev's inequality for random vectors is given. This inequality provides a lower bound for the percentage of the population of an arbitrary random vector X with finite mean μ = E(X) and a positive definite covariance matrix V = Cov(X) whose Mahalanobis distance with respect to V to the mean μ is less than a fixed value. The main advantage of the proof is that it is a simple exercise for a first year probability course. An alternative proof based on principal components is also provided. This proof can be used to study the case of a singular covariance matrix V.