Table of contents, Bernoulli Journal, vol. 9, no. 2 (2003)

Upper bounds and importance sampling of p-values for DNA and protein sequence alignments authors: Hock Peng Chan

GARCH processes: structure and estimation authors: Istv\'an Berkes , Lajos Horv\'ath , Piotr Kokoszka

Polynomial covariance functions on intervals authors: Sinjini Mitra , Tilmann Gneiting , Zolt\'an Sasv\'ari

Realized power variation and stochastic volatility models authors: Ole E. Barndorff-Nielsen , Neil Shephard

Empirical likelihood based hypothesis testing authors: John H.J. Einmahl , Ian W. Mckeague

Posterior consistency for semi-parametric regression problems authors: Messan Amewou-Atisso , Subhashis Ghosal , Jayanta K. Ghosh , R.V. Ramamoorthi

A survey and some generalizations of Bessel processes authors: Anja Göing-Jaeschke , Marc Yor

Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequence authors: Sven Erick Alm

Editorial board, Bernoulli Journal, vol. 9, no. 2 (2003)

Willem R. van Zwet,Editor-in-Chief

Sara A. van de Geer,Co-Editor

Peter J. Bickel,Editor

D.R. Cox,Editor

David Nualart,Editor

Edward C. Waymire,Editor

Adrian Baddeley,Associate Editor

Lucien Birgé,Associate Editor

Phillipe Carmona,Associate Editor

Paul Embrechts,Associate Editor

Evarist Giné,Associate Editor

Friedrich Götze,Associate Editor

Peter Guttorp,Associate Editor

Frank den Hollander,Associate Editor

Iain Johnstone,Associate Editor

Wilfrid S. Kendall,Associate Editor

M.R. Leadbetter,Associate Editor

Enno Mammen,Associate Editor

Peter McCullagh,Associate Editor

Thomas Mikosch,Associate Editor

Dominique Picard,Associate Editor

David Siegmund,Associate Editor

Simon Tavaré,Associate Editor

Dag Tjøstheim,Associate Editor

Tineke H. Bakker,Editorial Assistant

Richard Leigh,Copy Editor

Ank Lepping,Desk Editor

Ole E. Barndorff-Nielsen (1995-2000),Past Editor-in-Chief

Jens L. Jensen (1995-2000),Past Managing Co-editor

We show in general how the substitution matrix and gap penalty function for local sequence alignments can be chosen such that the score statistic grows at a logarithmic rate when the two sequences are unrelated. The method used is the construction of a mixture distribution in which sequences with large scores are generated with uniformly higher likelihood. This distribution is also used for the importance sampling of the $p$-value of the score. An upper bound of this $p$-value is computed and compared against the simulated value.

We study the structure of a GARCH$(p,q)$ sequence. We show that the conditional variance can be written as an infinite sum of the squares of the previous observations and that the representation is unique. We prove the consistency and asymptotic normality of the quasi-maximum likelihood estimator of the parameters of the GARCH$(p,q)$ sequence under mild conditions.

A characteriztion is presented of the class of stationary processes that have polynomial covariance functions of degree less than or equal to 4 on an interval. The results extend to isotropic random fields and have applications in spatial statistics.

Limit distribution results on realized power variation, that is, sums of absolute powers of increments of a process, are derived for certain types of semimartingale with continuous local martingale component, in particular for a class of flexible stochastic volatility models. The theory covers, for example, the cases of realized volatility and realized absolute variation. Such results should be helpful in, for example, the analysis of volatility models using high-frequency information.

Omnibus tests for various nonparametric hypotheses are developed using the empirical likelihood method. These include tests for symmetry about zero, changes in distribution, independence and exponentiality. The approach is to localize the empirical likelihood using a suitable `time' variable implicit in the null hypothesis and then form an integral of the log-likelihood ratio statistic. The asymptotic null distributions of these statistics are established. In simulation studies, the proposed statistics are found to have greater power than corresponding Cram\'er--von Mises type statistics.

We consider Bayesian inference in the linear regression problem with an unknown error distribution that is symmetric about zero. We show that if the prior for the error distribution assigns positive probabilities to a certain type of neighbourhood of the true distribution, then the posterior distribution is consistent in the weak topology. In particular, this implies that the posterior distribution of the regression parameters is consist\-ent in the Euclidean metric. The result follows from our generalization of a celebrated result of Schwartz to the independent, non-identical case and the existence of exponentially consistent tests of the complement of the neighbourhoods shown here. We then specialize to two important prior distributions, the Polya tree and Dirichlet mixtures, and show that under appropriate conditions these priors satisfy the positivity requirement of the prior probabilities of the neighbourhoods of the true density. We consider the case of both non-stochastic and stochastic regressors. A similar problem of Bayesian inference in a generalized linear model for binary responses with an unknown link is also considered.

Bessel processes play an important role in financial mathematics because of their strong relation to financial models such as geometric Brownian motion or Cox-Ingersoll-Ross processes. We are interested in the first time Bessel processes and, more generally, radial Ornstein-Uhlenbeck processes hit a given barrier. We give explicit expressions of the Laplace transforms of first hitting times by (squared) radial Ornstein-Uhlenbeck processes, that is, Cox-Ingersoll-Ross processes. As a natural extension we study squared Bessel processes and squared Ornstein-Uhlenbeck processes with negative dimensions or negative starting points and derive their properties.

For $n\ge 0$, let $\lambda_n$ be the median of the $\Gamma(n+1,1)$ distribution. We prove that the sequence $\{\alphapha_n=\lambda_n-n\}$ decreases from $\log 2$ to $\frac{2}{3}$ as $n$ increases from 0 to $\infty$. The difference, $1-\alphapha_n$, between the mean and the median thus increases from $1-\log 2$ to $\frac{1}{3}$. This result also proves a conjecture by Chen and Rubin about the Poisson distributions: if $Y_{\mu}\sim\text{Poisson}(\mu)$, and $\lambda_n$ is the largest $\mu$ such that $P(Y_{\mu}\le n)=\frac{1}{2}$, then $\lambda_n-n$ is decreasing in $n$. The sequence $\{\alphapha_n\}$ is related to a sequence $\{\theta_n\}$, introduced by Ramanujan, which is known to be decreasing and of the form $\theta_n=\frac{1}{3}+4/(135(n+k_n))$, where $\frac{2}{21}<k_n\le\frac{8}{45}$. We also show that the sequence $\{k_n\}$ is decreasing.