Comparing the relative efficiency of three sequential procedures

We consider the sequential procedures developed by Robbins and Siegmund (1974), Louis (1975) and Zoubeidi (1992) for comparing the means of two treatments. We let the procedures have equal power functions and compare their Bayes and minimax risks using the invariance property of their power functions. For each of several formulations of the problem we determine the most relatively efficient procedure and compute its expected total sample size.     

Acute Middle Ear Infection in Small Children: A Bayesian Analysis Using Multiple Time Scales

The study is based on a sample of 965 children living in Oulu region (Finland), who were monitored for acute middle ear infections from birth to the age of two years. We introduce a nonparametrically defined intensity model for ear infections, which involves both fixed and time dependent covariates, such as calendar time, current age, length of breast-feeding time until present, or current type of day care. Unmeasured heterogeneity, which manifests itself in frequent infections in some children and rare in others and which cannot be explained in terms of the known covariates, is modelled by using individual frailty parameters. A Bayesian approach is proposed to solve the inferential problem. The numerical work is carried out by Monte Carlo integration (Metropolis-Hastings algorithm).     

Estimation of Dynamic Bivariate Mixture Models

This note compares a Bayesian Markov chain Monte Carlo approach implemented by Watanabe with a maximum likelihood ML approach based on an efficient importance sampling procedure to estimate dynamic bivariate mixture models. In these models, stock price volatility and trading volume are jointly directed by the unobservable number of price-relevant information arrivals, which is specified as a serially correlated random variable. It is shown that the efficient importance sampling technique is extremely accurate and that it produces results that differ significantly from those reported by Watanabe.     

Bayesian estimation for the Pareto income distribution

The Bayes estimators of the Gini index, the mean income and the proportion of the population living below a prescribed income level are obtained in this paper on the basis of censored income data from a pareto income distribution. The said estimators are obtained under the assumptions of a two-parameter exponential prior distribution and the usual squared error loss function. This work is also extended to the case when the income data are grouped and the exact incomes for the individuals in the population are not available. The method for the assessment of the hyperparameters is also outlined. Finally, the results are generalized for the doubly truncated gamma prior distribution. Now deceased.     

Bayesian quadrature with non-normal approximating functions

We consider an efficient Bayesian approach to estimating integration-based posterior summaries from a separate Bayesian application. In Bayesian quadrature we model an intractable posterior density function f(·) as a Gaussian process, using an approximating function g(·), and find a posterior distribution for the integral of f(·), conditional on a few evaluations of f (·) at selected design points. Bayesian quadrature using normal g (·) is called Bayes-Hermite quadrature. We extend this theory by allowing g(·) to be chosen from two wider classes of functions. One is a family of skew densities and the other is the family of finite mixtures of normal densities. For the family of skew densities we describe an iterative updating procedure to select the most suitable approximation and apply the method to two simulated posterior density functions.     

Properties of Bayes Factors Based on Test Statistics

Abstract.  This article examines the consistency, interpretation and application of Bayes factors constructed from standard test statistics. Primary conclusions are that Bayes factors based on multinomial and normal test statistics are consistent for suitable choices of the hyperparameters used to specify alternative hypotheses, and that such constructions can be extended to obtain consistent Bayes factors based on likelihood ratio statistics. A connection between Bayes factors based on likelihood ratio statistics and the Bayesian information criterion is exposed, as is a connection between Bayes factors based on F statistics and parametric Bayes factors based on normal-inverse gamma models. Similarly, Bayes factors based on chi-squared statistics for multinomial data are shown to provide accurate approximations to Bayes factors based on multinomial/Dirichlet models. An illustration of how the simple form of these Bayes factors can be exploited to generate easily interpretable summaries of the experimental 'weight of evidence' is provided.     

Wavelet Shrinkage with Double Weibull Prior

In this article, we propose a denoising methodology in the wavelet domain based on a Bayesian hierarchical model using Double Weibull prior. We propose two estimators, one based on posterior mean (Double Weibull Wavelet Shrinker, DWWS) and the other based on larger posterior mode (DWWS-LPM), and show how to calculate them efficiently. Traditionally, mixture priors have been used for modeling sparse wavelet coefficients. The interesting feature of this article is the use of non-mixture prior. We show that the methodology provides good denoising performance, comparable even to state-of-the-art methods that use mixture priors and empirical Bayes setting of hyperparameters, which is demonstrated by extensive simulations on standardly used test functions. An application to real-word dataset is also considered.     

Estimators for the Horvitz-Thompson Statistic Based on Some Posterior Distributions

The knowledge of first-order inclusion probabilities characterizing a sampling scheme is essential in design-based estimation of finite population totals. Sometimes the scheme is so complex that these probabilities cannot be computed exactly. Instead, both inclusion probabilities and corresponding sampling weights are simulated. One empirical Horvitz-Thompson estimator for a population total using simulation-based range-preserving estimates of sampling weights is obtained by applying the restricted maximum likelihood principle directly to each inclusion probability. The assumption of a prior distribution and the assessment of resulting posterior for a weight lead to two other estimators. One of them is the posterior mean estimator of the Horvitz-Thompson statistic. In a simulation involving Polish agricultural census data and a sequential fixed-cost sampling scheme, this estimator has attractive properties also from a frequentist point of view.     

Bayesian analysis of threshold autoregressions

A nonasymptotic Bayesian approach is developed for analysis of data from threshold autoregressive processes with two regimes. Using the conditional likelihood function, the marginal posterior distribution for each of the parameters is derived along with posterior means and variances. A test for linear functions of the autoregressive coefficients is presented. The approach presented uses a posterior p-value averaged over the values of the threshold. The one-step ahead predictive distribution is derived along with the predictive mean and variance. In addition, equivalent results are derived conditional upon a value of the threshold. A numerical example is presented to illustrate the approach.     

Probability matching priors for two-sided tolerance intervals in balanced one-way and two-way nested random effects models

We consider two-sided Bayesian tolerance intervals, with approximate frequentist validity, for a future observation in balanced one-way and two-way nested random effects models. Probability matching conditions, specific to this problem, are derived in either case via a technique that involves inversion of approximate posterior characteristic functions. In addition to yielding probability matching priors for the present problem, these conditions are useful in evaluating certain other priors that have received attention in the literature.