Table of contents, Bernoulli Journal, vol. 5, no. 5 (1999)

The concept of duality and applications to Markov processes arising in neutral population genetics models; Authors: Martin Möhle

Taylor expansions of curve-crossing probabilities; Authors: David G. Hobson, David Williams, Andrew T.A. Wood

Point stationarity in d dimensions and Palm theory; Authors: Hermann Thorisson

Consistency and strong inconsistency of group-invariant predictive inferences; Authors: Morris L. Eaton, William D. Sudderth

Parameter estimation for discretely observed stochastic volatility models; Authors: Valentine Genon-Catalot, Thierry Jeantheau, Catherine Laredo

Nonlinear wavelet estimation of time-varying autoregressive processes; Authors: Rainer Dahlhaus, Michael H. Neumann, Rainer Von Sachs

On pointwise adaptive nonparametric deconvolution; Authors: Alexander Goldenshluger

Nonparametric estimation of quadratic regression functionals; Authors: Li-Shan Huang, Jianqing Fan

Editorial board, Bernoulli Journal, vol. 5, no. 5 (1999)

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

Xiao-Li Meng, Associate Editor

Thomas Mikosch, Associate Editor

Dominique Picard, Associate Editor

Yaacov Ritov, Associate Editor

David Siegmund, Associate Editor

Dag Tjøstheim, Associate Editor

Sara van de Geer, Associate Editor

Alan Welsh, 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)

One possible and widely used definition of the duality of Markov processes employs functions H relating one process to another in a certain way. For given processes X and Y the space U of all such functions H, called the duality space of X and Y, is studied in this paper. The algebraic structure of U is closely related to the eigenvalues and eigenvectors of the transition matrices of X and Y. Often as for example in physics (interacting particle systems) and in biology (population genetics models) dual processes arise naturally by looking forwards and backwards in time. In particular, time-reversible Markov processes are self-dual. In this paper, results on the duality space are presented for classes of haploid and two-sex population models. For example dim U = N+ 3 for the classical haploid Wright-Fisher model with fixed population size N.

Using an approach based on the Cameron-Martin-Girsanov theorem, we obtain a Taylor expansion for the probability that Brownian motion hits a smooth nonlinear boundary which grows at a suitable rate. The structure and probabilistic meaning of the terms in the expansion are studied in some detail.

The paper extends to d > 1 dimensions the concept of point-stationarity, which formalizes the intuitive idea of a point process for which the behaviour relative to a given point of the process is independent of the point selected as origin. After defining point-stationarity, this concept is characterized in several ways and the characterizations then used to extend to d dimensions a particular approach to Palm theory, producing two dualities between stationary and point-stationary processes with quite different interpretations. The dualities coincide in the ergodic case.

Consider a statistical model which is invariant under a group of transformations that acts transitively on the parameter space. In this situation, the problem of constructing invariant predictive distributions is considered. It is shown, under certain assumptions, that Fisherian pivoting and the use of right Haar measure as an improper prior distribution both yield the same invariant predictive distribution. Furthermore, it is shown that any other invariant predictive distribution is strongly inconsistent in the sense of Stone.

This paper deals with parameter estimation for stochastic volatility models. We consider a two-dimensional diffusion process (Y_t,V_t). Only (Y_t) is observed at n discrete times with a regular sampling interval. The unobserved coordinate (V_t) rules the diffusion coefficient (volatility) of (Y_t) and is an ergodic diffusion depending on unknown parameters. We build estimators of the parameters present in the stationary distribution of (V_t), based on appropriate functions of the observations. Consistency is proved under the asymptotic framework that the sampling interval tends to 0, while the number of observations and the length of the observation time tend to infinity. Asymptotic normality is obtained under an additional condition on the rate of convergence of the sampling interval. Examples of models from finance are treated, and numerical simulation results are given.

We consider nonparametric estimation of the parameter functions a_i(.), i = 1,...,p, of a time-varying autoregressive process. Choosing an orthonormal wavelet basis representation of the functions a_i, the empirical wavelet coefficients are derived from the time series data as the solution of a least-squares minimization problem. In order to allow the a_i to functions of inhomogeneous regularity, we apply nonlinear thresholding to the empirical coefficients and obtain locally smoothed estimates of the a_i. We show that the resulting estimators attain the usual minimax L₂ rates up to a logarithmic factor, simultaneously in a large scale of Besov classes. The finite-sample behaviour of our procedure is demonstrated by application to two typical simulated examples.

We consider estimating an unknown function f from indirect white noise observations with particular emphasis on the problem of nonparametric deconvolution. Nonparametric estimators that can adapt to unknown smoothness of f are developed. The adaptive estimators are specified under two sets of assumptions on the kernel of the convolution transform. In particular, kernels having the Fourier transform with polynomially and exponentially decaying tails are considered. It is shown that the proposed estimates possess, in a sense, the best possible abilities for pointwise adaptation.

Quadratic regression functionals are important for bandwidth selection of nonparametric regression techniques and for nonparametric goodness-of-fit test. Based on local polynomial regression, we propose estimators for weighted integrals of squared derivatives of regression functions. The rates of convergence in mean square error are calculated under various degrees of smoothness and appropriate values of the smoothing parameter. Asymptotic distributions of the proposed quadratic estimators are considered with the Gaussian noise assumption. It is shown that when the estimators are pseudo-quadratic (linear components dominate quadratic components), asymptotic normality with rate n^-1/2 can be achieved.