A note on the central limit theorem for the bootstrap mean

Following Yang (1988), a simple, more self-contained derivation of the asymptotic normality of the bootstrap mean is presented, as well as other asymptotic results. The derivations are appropriate for beginning graduate students in statistics, relying only on fundamental notions of probability theory and analysis,     

Application of extreme value theory to the bec family of distributions

Arnold and Strauss (1988) derived a family of bivariate life distributions having the property that the conditional distributions are exponential. Asymptotic distributions for the marginal and bivariate extremes for this family of distributions are derived employing the asymptotic theory of extreme order statistics.     

A powerful test for uniformity to detect highly peaked alternativedensities

A new class of statistical tests for uniformity based on sample ranges is proposed to detect a uniform density contaminated by a density with one or more high peaks. These kinds of alternatives occur quite frequently, especially in physics. It is shown that the proposed tests arc consistent and have high Pitman asymptotic relative efficiencies. Results of a Monte Carlo power study indicate that they, compared with other tests of uniformity, possess good power properties.A comparative study of various tests is also conducted using real data. An effcient algorithm for computing out test statistic and a table of percentage points are given, providing a practical guide for using the new test.     

Gibbs sampling for marginal posterior expectations

In earlier work (Gelfand and Smith, 1990 and Gelfand et al, 1990) a sampling based approach using the Gibbs sampler was offered as a means for developing marginal posterior densities for a wide range of Bayesian problems several of which were previously inaccessible. Our purpose here is two-fold. First we flesh out the implementation of this approach for calculation of arbitrary expectations of interest. Secondly we offer comparison with perhaps the most prominent approach for calculating posterior expectations, analytic approximation involving application of the LaPlace method. Several illustrative examples are discussed as well. Clear advantages for the sampling based approach emerge.     

Evaluation of an analysis approach used to account for extra-variation in clustered categorical responses

This article presents the results of a simulation study investigating the performance of an approach developed by Miller and Landis (1991) for the analysis of clustered categorical responses. Evaluation of this “two-step” approach, which utilizes the method of moments to estimate the extra-variation pardmeters and subsequently incorporates these parameters into estimating equations for modelling the marginal expectations, is carried out in an experimental setting involving a comparison between two groups of observations. We assume that data for both groups are collected from each cluster and responses are measured on a three-point ordinal scale. The performance of the estimators used in both “steps” of the analysisis investigated and comparisons are made to an alternative analysismethod that ignores the clustering. The results indicate that in the chosen setting the test for a difference between groups generally operatbs at the nominal α=0.05 for 10 or more clusters and hasincreasing power with both an increasing number of clusters and an inrreasing treatment effect. These results provide a striking contrasc to those obtained from an improper analysis that ignores clustering.     

Correlation in a Bayesian framework

The authors consider the correlation between two arbitrary functions of the data and a parameter when the parameter is regarded as a random variable with given prior distribution. They show how to compute such a correlation and use closed form expressions to assess the dependence between parameters and various classical or robust estimators thereof, as well as between p‐values and posterior probabilities of the null hypothesis in the one‐sided testing problem. Other applications involve the Dirichlet process and stationary Gaussian processes. Using this approach, the authors also derive a general nonparametric upper bound on Bayes risks.     

Link-tracing sampling with an initial sequential sample of sites: Estimating the size of a hidden human population

We proposed a modification to the variant of link-tracing sampling suggested by Félix-Medina and Thompson [M.H. Félix-Medina, S.K. Thompson, Combining cluster sampling and link-tracing sampling to estimate the size of hidden populations, Journal of Official Statistics 20 (2004) 19–38] that allows the researcher to have certain control of the final sample size, precision of the estimates or other characteristics of the sample that the researcher is interested in controlling. We achieve this goal by selecting an initial sequential sample of sites instead of an initial simple random sample of sites as those authors suggested. We estimate the population size by means of the maximum likelihood estimators suggested by the above-mentioned authors or by the Bayesian estimators proposed by Félix-Medina and Monjardin [M.H. Félix-Medina, P.E. Monjardin, Combining link-tracing sampling and cluster sampling to estimate the size of hidden populations: A Bayesian-assisted approach, Survey Methodology 32 (2006) 187–195]. Variances are estimated by means of jackknife and bootstrap estimators as well as by the delta estimators proposed in the two above-mentioned papers. Interval estimates of the population size are obtained by means of Wald and bootstrap confidence intervals. The results of an exploratory simulation study indicate good performance of the proposed sampling strategy.     

Data-driven tests of uniformity on product manifolds

Data-driven versions of Sobolev tests of uniformity on compact Riemannian manifolds are reviewed and their large-sample asymptotic properties are given. A variant which is suitable for product manifolds is introduced. Data-driven goodness-of-fit tests of multivariate distributions are derived from data-driven tests of uniformity on tori.