Table of contents, Bernoulli Journal, vol. 6, no. 2 (2000)

Tail probabilities for the null distribution of scanning statistics; Authors: David Siegmund, Benjamin Yakir

Estimating the second largest eigenvalue of a Markov transition matrix; Authors: Steven T. Garren, Richard L. Smith

Likelihood-based inference with singular information matrix; Authors: Andrea Rotnitzky, David R. Cox, Matteo Bottai, James Robins

Resampling and exchangeable arrays; Authors: Peter Mccullagh

Rare events simulation for heavy-tailed distributions; Authors: Søren Asmussen, Klemens Binswanger, Bjarne Højgaard

Change of measures for Markov chains and the LlogL theorem for branching processes; Authors: Krishna B. Athreya

Large deviations for stochastic Volterra equations; Authors: David Nualart, Carles Rovira

Approximation of the Ornstein-Uhlenbeck local time by harmonic oscillators; Authors: José R. León, Gonzalo Perera

Editorial board, Bernoulli Journal, vol. 6, no. 2 (2000)

Ole E. Barndorff-Nielsen, Editor-in-Chief

Jens Ledet Jensen, Managing Co-Editor

Peter J. Bickel, Editor

David Nualart, Editor

David R. Cox, Editor

Edward C. Waymire, Editor

Søren Asmussen, Associate Editor

Adrian Baddeley, Associate Editor

Lucien Birgé, Associate Editor

Philippe Carmona, Associate Editor

Louis H. Y. Chen, Associate Editor

Hans Föllmer, Associate Editor

Valentine Genon-Catalot, Associate Editor

Richard D. Gill, Associate Editor

Peter Guttorp, Associate Editor

Peter McCullagh, Associate Editor

Alexander A. Novikov, Associate Editor

Yosihiko Ogata, Associate Editor

Victor Pérez-Abreu, Associate Editor

Andrea Rotnitzky, Associate Editor

Terence P. Speed, Associate Editor

Dag Tjøstheim, Associate Editor

Richard L. Tweedie, Associate Editor

Andrew T.A. Wood, Associate Editor

Ank Lepping, Desk Editor

This paper is concerned with statistics that scan a multidimensional spatial region to detect a signal against a noisy background. The background is modelled as independent observations from an exponential family of distributions with a known `null' value of the natural parameter, while the signal is given by independent observations from the same exponential family, but with a different value of the parameter on a particular subregion of the spatial domain. The main result is an extension to multidimensional time of the method of Pollak and Yakir, which relies on a change of measure motivated by change-point analysis, to evaluate approximately the null distribution of the likelihood ratio statistic. Both large-deviation and Poisson approximations are obtained.

The number of iterations required to estimate accurately the stationary distribution of a Markov chain is determined by a preliminary sample to estimate the convergence rate, which is related to the second largest eigenvalue of the transition operator. The estimator of the second largest eigenvalue, along with those of two nuisance parameters, can be shown to converge to their true values in probability, and a form of the central limit theorem is proved. Explicit expressions for the bias and variance of the asymptotic distribution of this estimator are derived. A theoretical standard is derived against which other estimators of the second largest eigenvalue may be judged. An application is given involving the use of the Gibbs sampler to calculate a posterior distribution.

We consider likelihood-based asymptotic inference for a p-dimensional parameter θ of an identifiable parametric model with singular information matrix of rank p-1 at θ=θ^* and likelihood differentiable up to a specific order. We derive the asymptotic distribution of the likelihood ratio test statistics for the simple null hypothesis that θ=θ^* and of the maximum likelihood estimator (MLE) of θ when θ=θ^*. We show that there exists a reparametrization such that the MLE of the last p-1 components of θ converges at rate$O_p\left(n^{-1/2}\right)$ . For the first component θ₁ of θ the rate of convergence depends on the order s of the first non-zero partial derivative of the log-likelihood with respect to θ₁ evaluated at θ^*. When s is odd the rate of convergence of the MLE of θ₁ is$O_p\left(n^{-1/2s}\right)$ . When s is even, the rate of convergence of the MLE of$|{\theta }_1-{\theta }_1^{*}|$ is$O_p\left(n^{-1/2s}\right)$ and, moreover, the asymptotic distribution of the sign of the MLE of θ₁-θ₁^* is non-standard. When p=1 it is determined by the sign of the sum of the residuals from the population least-squares regression of the (s+1)th derivative of the individual contributions to the log-likelihood on their derivatives of order s. For p>1, it is determined by a linear combination of the sum of residuals of a multivariate population least-squares regression involving partial and mixed derivatives of the log-likelihood of a specific order. Thus although the MLE of |θ₁-θ₁^*| has a uniform rate of convergence of$O_p\left(n^{-1/2s}\right)$ , the uniform convergence rate for the MLE of θ₁ in suitable shrinking neighbourhoods of θ₁^* is only$O_p\left(n^{-1/(2s+2)}\right)$ .

The nonparametric, or resampling, bootstrap for a single unstructured sample corresponds to the algebraic operation of monoid composition, with a uniform distribution on the monoid. With this interpretation, the notion of resampling can be extended to designs having a certain group-invariance property. Two types of exchangeable array structures are considered in some detail, namely the one-way layout, and the two-way row-column exchangeable design. Although in both cases there is a unique group under which the sampling distribution of the observations is exchangeable, the choice of monoid is not unique. Different choices of monoid can lead to drastically different, and in some cases quite misleading, inferences.

This paper studies rare events simulation for the heavy-tailed case, where some of the underlying distributions fail to have the exponential moments required for the standard algorithms for the light-tailed case. Several counterexamples are given to indicate that in the heavy-tailed case, there are severe problems with the approach of developing limit results for the conditional distribution given the rare event; this is used as a basis for importance sampling. On the positive side, two algorithms having a relative error which is almost bounded are presented, one based upon order statistics and the other upon a different importance sampling idea.

Let P(.,.) be a probability transition function on a measurable space$(M,boldM)$ . Let V(.) be a strictly positive eigenfunction of P with eigenvalue ρ>0. Let $$\tilde{P}(x,dy)\equiv \frac{V\left(y\right)P(x,dy)}{\rho V\left(x\right)}.$$ Then$\tilde{P}(.,.)$ is also a transition function. Let P_x and${\tilde{P}}_x$ denote respectively the probability distribution of a Markov chain$\{X_j{\}}_0^{\mathrm{\infty }}$ with X₀=x and transition functions P and$\tilde{P}$ . Conditions for${\tilde{P}}_x$ to be dominated by P_x or to be singular with respect to P_x are given in terms of the martingale sequence$W_n\equiv V\left(X_n\right)/{\rho }^n$ and its limit. This is applied to establish an LlogL theorem for supercritical branching processes with an arbitrary type space.

This paper is devoted to prove a large-deviation principle for solutions to multidimensional stochastic Volterra equations.

We consider a particle of mass 1/β submitted to the action of an harmonic oscillator. If we add a white-noise external force, it is well known that the trajectories of the particle, for β tending to infinity, converge to an Ornstein-Uhlenbeck process. Using the number of crossings of the particle with a fixed level u, we construct a consistent estimator of the Ornstein-Uhlenbeck local time, giving an estimate of the speed of this convergence.