Table of contents, Bernoulli Journal, vol. 11, no. 5 (2005)

Nonparametric inference for Lévy-driven Ornstein-Uhlenbeck processes; Authors: G. Jongbloed, F.H. Van Der Meulen, A.W. Van Der Vaart

Distance-reducing Markov bases for sampling from a discrete sample space; Authors: Akimichi Takemura, Satoshi Aoki

On adaptive Markov chain Monte Carlo algorithms; Authors: Yves F. Atchadé, Jeffrey S. Rosenthal

Central limit theorem and convergence to stable laws in Mallows distance; Authors: Oliver Johnson, Richard Samworth

Binary sequential representations of random partitions; Authors: James E. Young

An asymptotic theory for the nonparametric maximum likelihood estimator in the Cox gene model; Authors: I-Shou Chang, Chao Agnes Hsiung, Mei-Chuan Wang, Chi-Chung Wen

Optimal quantization methods for nonlinear filtering with discrete-time observations; Authors: Gilles Pagès, Huyên Pham

On invariant distribution function estimation for continuous-time stationary processes; Authors: Dominique Dehay

Editorial board, Bernoulli Journal, vol. 11, no. 5 (2005)

Peter McCullagh, Editor-in-Chief

Peter J. Bickel, Editor

D.R. Cox, Editor

David Nualart, Editor

Edward Waymire, Editor

Lucien Birgé, Associate Editor

Evarist Giné, Associate Editor

Peter Hall, Associate Editor

Wilfrid S. Kendall, Associate Editor

Thomas Mikosch, Associate Editor

Gesine Reinert, Associate Editor

Ya'acov Ritov, Associate Editor

David Siegmund, Associate Editor

Dag Tjøstheim, Associate Editor

Alexandre Tsybakov, Associate Editor

Sara van de Geer, Associate Editor

Eva Bjorn Vedel Jensen, Associate Editor

Alan Welsh, Associate Editor

Wei-Biao Wu, Associate Editor

Mitzi Nakatsuka, Editorial Assistant

Richard Leigh, Copy Editor

Ank Lepping, Desk Editor

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

Willem R. van Zwet, Past Editors-in-Chief(2000-2003)

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

Sara van de Geer, Past Co-Editors (2000-2003)

We consider nonparametric estimation of the Lévy measure of a hidden Lévy process driving a stationary Ornstein-Uhlenbeck process which is observed at discrete time points. This Lévy measure can be expressed in terms of the canonical function of the stationary distribution of the Ornstein-Uhlenbeck process, which is known to be self-decomposable. We propose an estimator for this canonical function based on a preliminary estimator of the characteristic function of the stationary distribution. We provide a support-reduction algorithm for the numerical computation of the estimator, and show that the estimator is asymptotically consistent under various sampling schemes. We also define a simple consistent estimator of the intensity parameter of the process. Along the way, a nonparametric procedure for estimating a self-decomposable density function is constructed, and it is shown that the Ornstein-Uhlenbeck process is β-mixing. Some general results on uniform convergence of random characteristic functions are included.

We study Markov bases for sampling from a discrete sample space equipped with a convenient metric. Starting from any two states in the sample space, we ask whether we can always move closer by an element of a Markov basis. We call a Markov basis distance-reducing if this is the case. The particular metric we consider in this paper is the L₁-norm on the sample space. Some characterizations of L₁-norm-reducing Markov bases are derived.

We look at adaptive Markov chain Monte Carlo algorithms that generate stochastic processes based on sequences of transition kernels, where each transition kernel is allowed to depend on the history of the process. We show under certain conditions that the stochastic process generated is ergodic, with appropriate stationary distribution. We use this result to analyse an adaptive version of the random walk Metropolis algorithm where the scale parameter σ is sequentially adapted using a Robbins-Monro type algorithm in order to find the optimal scale parameter σ_opt. We close with a simulation example.

We give a new proof of the classical central limit theorem, in the Mallows (L^r-Wasserstein) distance. Our proof is elementary in the sense that it does not require complex analysis, but rather makes use of a simple subadditive inequality related to this metric. The key is to analyse the case where equality holds. We provide some results concerning rates of convergence. We also consider convergence to stable distributions, and obtain a bound on the rate of such convergence.

Random partitions can be thought of as a consistent family of exchangeable random partitions of the sets {1,2,...,n} for n≥1. Historically, random partitions were constructed by sampling an infinite population of types and partitioning individuals of the same type into a single class. A particularly tractable way to construct random partitions is via random sequences of 0s and 1s. The only random partition derived from an independent 0-1 sequence is Ewens' one-parameter family of partitions which plays a predominant role in population genetics. A two-parameter generalization of Ewens' partition is obtained by considering random partitions constructed from discrete renewal processes and introducing a convolution-type product on 0-1 sequences.

The Cox model with a gene effect for age at onset was introduced and studied by Li, Thompson and Wijsman. We study the nonparametric maximum likelihood estimation of the gene effect and the regression coefficient in this model. We indicate conditions under which the parameters are identifiable and the nonparametric maximum likelihood estimate is consistent and asymptotically normal. We also apply the theory of observed profile information to obtain a consistent estimate of the asymptotic variance. Besides providing theoretical support for Li et al., our work provides an alternative approach to the numerical methods in this model.

We develop an optimal quantization approach for numerically solving nonlinear filtering problems associated with discrete-time or continuous-time state processes and discrete-time observations. Two quantization methods are discussed: a marginal quantization and a Markovian quantization of the signal process. The approximate filters are explicitly solved by a finite-dimensional forward procedure. A posteriori error bounds are stated, and we show that the approximate error terms are minimal at some specific grids that may be computed off-line by a stochastic gradient method based on Monte Carlo simulations. Some numerical experiments are carried out: the convergence of the approximate filter as the accuracy of the quantization increases and its stability when the latent process is mixing are emphasized.

This paper is concerned with the asymptotic behaviour of the empirical distribution function for a large class of continuous-time weakly dependent stationary processes. Under mild mixing conditions the empirical distribution function is an unbiased consistent estimator of the marginal distribution function of the process. For strongly mixing processes this estimator is asymptotically normal. We propose a consistent estimator of the asymptotic variance, and then study the functional central limit theorem for the empirical distribution function.