Table of contents, Bernoulli Journal, vol. 8, no. 4 (2002)

Nonlinear kernel density estimation for binned data: convergence in entropy; Authors: Gordon Blower, Julia E. Kelsall

Non-informative priors in the case of Gaussian long-memory processes; Authors: Anne Philippe, Judith Rousseau

Application of structural risk minimization to multivariate smoothing spline regression estimates; Authors: Michael Kohler, Adam Krzyzak, Dominik Schäfer

Estimation of diffusion parameters for discretely observed diffusion processes; Authors: Helle Sørensen

Making Markov martingales meet marginals: with explicit constructions; Authors: Dilip B. Madan, Marc Yor

A weak criterion of absolute continuity for jump processes: application to the Boltzmann equation; Authors: Nicolas Fournier, Sylvie Méléard

Editorial board, Bernoulli Journal, vol. 8, no. 4 (2002)

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)

A method is proposed for creating a smooth kernel density estimate from a sample of binned data. Simulations indicate that this method produces an estimate for relatively finely binned data which is close to what one would obtain using the original unbinned data. The kernel density estimate $\hat {f}$, is the stationary distribution of a Markov process resembling the Ornstein-Uhlenbeck process. This $\hat {f}$, may be found by an iteration scheme which converges at a geometric rate in the entropy pseudo-metric, and hence in $L^1$, and transportation metrics. The proof uses a logarithmic Sobolev inequality comparing relative Shannon entropy and relative Fisher information with respect to $\hat {f}$.

In this paper, we consider an asymptotic Bayesian analysis for Gaussian processes with long memory. First, we determine the asymptotic expansion of the posterior density based on a normal approximation. This expansion leads to the construction of Bayesian confidence regions such as highest posterior density regions and to the determination of matching prior. Then, we generalize Clarke and Barron's result in the long-memory set-up. More precisely, we establish the asymptotic expansion of the Kullback-Leibler distance between the true density and the marginal density of the observations. As in the independent and identically distributed case, this result gives an asymptotic justification of Berger and Bernardo's algorithm to obtain reference priors.

Estimation of regression functions from bounded, independent and identically distributed data is considered. Motivated by Vapnik's principle of structural risk minimization, a data-dependent choice of the smoothing parameter of multivariate smoothing spline estimates is proposed. The corresponding smoothing spline estimates automatically adapt to the unknown smoothness of the regression function and their $L^2$ errors achieve the optimal rate of convergence up to a logarithmic factor. The result is valid without any regularity conditions on the distribution of the design.

We study the estimation of diffusion parameters for one-dimensional, ergodic diffusion processes that are discretely observed. We discuss a method based on a functional relationship between the drift function, the diffusion function and the invariant density and use empirical process theory to show that the estimator is $\sqrt{n}$-consistent and in certain cases weakly convergent. The Chan-Karolyi-Longstaff-Sanders (CKLS) model is used as an example and a numerical example is presented.

We present three generic constructions of martingales that all have the Markov property with known and prespecified marginal densities. These constructions are further investigated for the special case when the prespecified marginals satisfy the scaling property and hence the only datum needed for the construction is the density at unit time. Interesting relations with stochastic orders are presented, along with numerous examples of the resulting martingales.

We first prove a general and quite simple criterion of absolute continuity, based on the use of almost sure derivatives, which is applicable even when integration by parts may not be used. We apply it to Poisson-driven stochastic differential equations. Next, using a typically probabilistic substitution in the Boltzmann equation, we extend Tanaka's probabilistic interpretation for spatially homogeneous Boltzmann equations with Maxwell molecules and without angular cut-off to much more general spatially homogeneous two-dimensional Boltzmann equations. We relate a measure-solution {Q_t}_t of the equation to a solution V_t of a nonlinear Poisson-driven stochastic differential equation: for each t, Q_t is the law of V_t. We extend our absolute continuity criterion to these nonlinear Poisson functionals and prove that even in the case of degenerate initial distribution, the law of V_t admits a density f(t,⋅) for each t>0, which is hence a solution to the Boltzmann equation. We thus obtain an original existence result.