Table of contents, Bernoulli Journal, vol. 6, no. 5 (2000)

Inhomogeneous Markov point processes by transformation; Authors: Eva B. Vedel Jensen, Linda Stougaard Nielsen

Time-invariance estimating equations; Authors: A.J. Baddeley

Bootstrap relative errors and sub-exponential distributions; Authors: Andrew T.A. Wood

Distribution and dependence-function estimation for bivariate extreme-value distributions; Authors: Peter Hall, Nader Tajvidi

The multiple change-points problem for the spectral distribution; Authors: Marc Lavielle, Carenne Ludeña

On local times of a symmetric stable process as a doubly indexed process; Authors: Nathalie Eisenbaum

Approximation and support theorem for a wave equation in two space dimensions; Authors: Annie Millet, Marta Sanz-Solé

Z-theorems: limits of stochastic equations; Authors: Vladimir V. Anisimov, Georg CH. Pflug

Krein condition in probabilistic moment problems; Authors: Jordan Stoyanov

Editorial board, Bernoulli Journal, vol. 6, no. 5 (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

We construct parametrized models for point processes, allowing for both inhomogeneity and interaction. The inhomogeneity is obtained by applying parametrized transformations to homogeneous Markov point processes. An interesting model class, which can be constructed by this transformation approach, is that of exponential inhomogeneous Markov point processes. Statistical inference for such processes is discussed in some detail.

We describe a general method for deriving estimators of the parameter of a statistical model, with particular relevance to highly structured stochastic systems such as spatial random processes and `graphical' conditional independence models. The method is based on representing the stochastic model $X$ as the equilibrium distribution of an auxiliary Markov process $Y=(Y_t,t>0)$ where the discrete or continuous 'time' index t is to be understood as a fictional extra dimension added to the original setting. The parameter estimate $\hat{\theta }$ is obtained by equating to zero the generator of $Y$ applied to a suitable statistic and evaluated at the data $x$ . This produces an unbiased estimating equation for θ. Natural special cases include maximum likelihood, the method of moments, the reduced sample estimator in survival analysis, the maximum pseudolikelihood estimator for random fields and for point processes, the Takacs-Fiksel method for point processes, 'variational' estimators for random fields and multivariate distributions, and many standard estimators in stochastic geometry. The approach has some affinity with the Stein-Chen method for distributional approximation.

For the purposes of this paper, a distribution is sub-exponential if it has finite variance but its moment generating function is infinite on at least one side of the origin. The principal aim here is to study the relative error properties of the bootstrap approximation to the true distribution function of the sample mean in the important sub-exponential cases. Our results provide a fairly general description of how the bootstrap approximation breaks down in the tail when the underlying distribution is sub-exponential and satisfies some very mild additional conditions. The broad conclusion we draw is that the accuracy of the bootstrap approximation in the tail depends, in a rather sensitive way, on the precise tail behaviour of the underlying distribution. Our results are applied to several sub-exponential distributions, including the lognormal. The lognormal case is of particular interest because, as the simulation studies of Lee and Young have shown, bootstrap confidence intervals can have very poor coverage accuracy when applied to data from the lognormal.

Two new methods are suggested for estimating the dependence function of a bivariate extreme-value distribution. One is based on a multiplicative modification of an earlier technique proposed by Pickands, and the other employs spline smoothing under constraints. Both produce estimators that satisfy all the conditions that define a dependence function, including convexity and the restriction that its curve lie within a certain triangular region. The first approach does not require selection of smoothing parameters; the second does, and for that purpose we suggest explicit tuning methods, one of them based on cross-validation.

We consider the problem of detecting an unknown number of change-points in the spectrum of a second-order stationary random process. To reach this goal, some maximal inequalities for quadratic forms are first given under very weak assumptions. In a parametric framework, and when the number of changes is known, the change-point instants and the parameter vector are estimated using the Whittle pseudo-likelihood of the observations. A penalized minimum contrast estimate is proposed when the number of changes is unknown. The statistical properties of these estimates hold for strongly mixing and also long-range dependent processes. Estimation in a nonparametric framework is also considered, by using the spectral measure function. We conclude with an application to electroencephalogram analysis.

We consider the local time process $(L_t^x,x\in ℝ,t\ge 0)$ of a symmetric stable process X with an index β in (1,2]. We compute the p-variation of L on any rectangle of $ℝ\times [0,\mathrm{\infty })$ . Unlike for the p-variation of L with respect to the spatial parameter (studied by Marcus and Rosen), we show here that the Brownian case - when β= 2 - is atypical.

We prove a characterization of the support of the law of the solution for a stochastic wave equation with two-dimensional space variable, driven by a noise white in time and correlated in space. The result is a consequence of an approximation theorem, in the convergence of probability, for equations obtained by smoothing the random noise. For some particular classes of coefficients, approximation in the L^p-norm for p≥1 is also proved.

Let f_n(θ,ω) be a sequence of stochastic processes which converge weakly to a limit process f₀(θ,ω). We show under some assumptions the weak inclusion of the solution sets ${\theta }_n\left(\omega \right)=\{\theta :f_n(\theta ,\omega )=0\}$ in the limiting solution set ${\theta }_0\left(\omega \right)=\{\theta :f_0(\theta ,\omega )=0\}$ . If the limiting solutions are almost surely singletons, then weak convergence holds. Results of this type are called Z-theorems (zero-theorems). Moreover, we give various more specific convergence results, which have applications for stochastic equations, statistical estimation and stochastic optimization.

In 1944 M.G. Krein proposed a condition throwing light on the moment problem for absolutely continuous probability distributions. This condition, implying non-uniqueness, is expressed in terms of a normalized logarithmic integral of the density and has different forms in the Hamburger moment problem (for distributions on the whole real line) and in the Stieltjes moment problem (for distributions on the positive real line). Other forms of the Krein condition, together with new conditions (smoothing and growth condition on the density) suggested by G.D. Lin and based on a work by H. Dym and H.P. McKean, led to a unique solution to the moment problem. We present new results, give new proofs of previously known results and discuss related topics.