Qi-Man Shao

Monte Carlo Estimation of Bayesian Credible and HPD Intervals

Abstract This article considers how to estimate Bayesian credible and highest probability density (HPD) intervals for parameters of interest and provides a simple Monte Carlo approach to approximate these Bayesian intervals when a sample of the relevant parameters can be generated from their respective marginal posterior distribution using a Markov chain Monte Carlo (MCMC) sampling algorithm. We also develop a Monte Carlo method to compute HPD intervals for the parameters of interest from the desired posterior distribution using a sample from an importance sampling distribution. We apply our methodology to a Bayesian hierarchical model that has a posterior density containing analytically intractable integrals that depend on the (hyper) parameters. We further show that our methods are useful not only for calculating the HPD intervals for the parameters of interest but also for computing the HPD intervals for functions of the parameters. Necessary theory is developed and illustrative examples—including a si...

Gaussian processes: inequalities, small ball probabilities and applications

Publisher Summary This chapter focuses on the inequalities, small ball probabilities, and application of Gaussian processes. It is well-known that the large deviation result plays a fundamental role in studying the upper limits of Gaussian processes, such as the Strassen type law of the iterated logarithm. However, the complexity of the small ball estimate is well-known, and there are only a few Gaussian measures for which the small ball probability can be determined completely. The small ball probability is a key step in studying the lower limits of the Gaussian process. It has been found that the small ball estimate has close connections with various approximation quantities of compact sets and operators, and has a variety of applications in studies of Hausdorff dimensions, rate of convergence in Strassens law of the iterated logarithm, and empirical processes.

Normal approximation by Stein's method

Preface.- 1.Introduction.- 2.Fundamentals of Steins Method.- 3.Berry-Esseen Bounds for Independent Random Variables.- 4.L^1 Bounds.- 5.L^1 by Bounded Couplings.- 6 L^1: Applications.- 7.Non-uniform Bounds for Independent Random Variables.- 8.Uniform and Non-uniform Bounds under Local Dependence.- 9.Uniform and Non-Uniform Bounds for Non-linear Statistics.- 10.Moderate Deviations.- 11.Multivariate Normal Approximation.- 12.Discretized normal approximation.- 13.Non-normal Approximation.- 14.Extensions.- References.- Author Index .- Subject Index.- Notation.

Normal approximation under local dependence

We establish both uniform and nonuniform error bounds of the Berry–Esseen type in normal approximation under local dependence. These results are of an order close to the best possible if not best possible. They are more general or sharper than many existing ones in the literature. The proofs couple Stein’s method with the concentration inequality approach.

On discriminating between long-range dependence and changes in mean

We develop a testing procedure for distinguishing between a long-range dependent time series and a weakly dependent time series with change-points in the mean. In the simplest case, under the null hypothesis the time series is weakly dependent with one change in mean at an unknown point, and under the alternative it is long-range dependent. We compute the CUSUM statistic T n , which allows us to construct an estimator k of a change-point. We then compute the statistic T n,1 based on the observations up to time k and the statistic T n,2 2 based on the observations after time k. The statistic M n = max[T n.1 , T n,2 ] converges to a well-known distribution under the null, but diverges to infinity if the observations exhibit long-range dependence. The theory is illustrated by examples and an application to the returns of the Dow Jones index.

Almost sure invariance principles for mixing sequences of random variables

An almost sure invariance principle for stationary mixing sequences of random variables with mean zero and finite variance is obtained when the mixing rate satisfies [Sigma]no(2n) 1. A similar result is also given under a higher moment condition.

A non-uniform Berry-Esseen bound via Stein's method

Abstract. This paper is part of our efforts to develop Steins method beyond uniform bounds in normal approximation. Our main result is a proof for a non-uniform Berry–Esseen bound for independent and not necessarily identically distributed random variables without assuming the existence of third moments. It is proved by combining truncation with Steins method and by taking the concentration inequality approach, improved and adapted for non-uniform bounds. To illustrate the technique, we give a proof for a uniform Berry–Esseen bound without assuming the existence of third moments.

The law of the iterated logarithm for negatively associated random variables

This paper proves that the law of the iterated logarithm holds for a stationary negatively associated sequence of random variables with finite variance. The proof is based on a Rosenthal type maximal inequality, a Kolmogorov type exponential inequality and Steins method.

A New Skewed Link Model for Dichotomous Quantal Response Data

Abstract The logit, probit, and student t-links are widely used in modeling dichotomous quantal response data. Most of the commonly used link functions are symmetric, except the complementary log-log link. However, in some applications the overall fit can be significantly improved by the use of an asymmetric link. In this article we propose a new skewed link model for analyzing binary response data with covariates. Introducing a skewed distribution for the underlying latent variable, we develop a class of asymmetric link models for binary response data. Using a Bayesian approach, we first characterize the propriety of the posterior distributions using standard improper priors. We further propose informative priors using historical data from a similar previous study. We examine the proposed method through a large-scale simulation study and use data from a prostate cancer study to demonstrate the use of historical data in Bayesian model fitting and comparison of skewed link models.

A note on the almost sure central limit theorem for weakly dependent random variables

We give here an almost sure central limit theorem for associated sequences, strongly mixing and p-mixing sequences under the same conditions that assure that the central limit theorem holds.