Table of contents, Bernoulli Journal, vol. 5, no. 2 (1999)

Estimation of Rényi exponents in random cascades; Authors: Brent M. Troutman, Aldo V. Vecchia

Random-design regression under long-range dependent errors; Authors: Sándor Csörgö, Jan Mielniczuk

The p-optimal martingale measure and its asymptotic relation with the minimal-entropy martingale measure; Authors: Peter Grandits

Laplace transforms related to excursions of a one-dimensional diffusion; Authors: Jim Pitman, Marc Yor

Expansion of the density: a Wiener-chaos approach; Authors: David Márquez-Carreras, Marta Sanz-Solé

The Hölder and the Besov regularity of the density for the solution of a parabolic stochastic partial differential equation; Authors: Pierre-Luc Morien

Estimating equations based on eigenfunctions for a discretely observed diffusion process; Authors: Mathieu Kessler, Michael Sørensen

Asymptotic normality of posterior distributions in high-dimensional linear models; Authors: Subhashis Ghosal

Minimax nonparametric hypothesis testing: the case of an inhomogeneous alternative; Authors: Oleg V. Lepski, Vladimir G. Spokoiny

Derivation of equivalent kernel for general spline smoothing: a systematic approach; Authors: Felix Abramovich, Vadim Grinshtein

Editorial board, Bernoulli Journal, vol. 5, no. 2 (1999)

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 consider statistical estimation of the Rényi exponent $\tau \left(h\right)$, which characterizes the scaling behaviour of a singular measure $\mu$ defined on a subset of $R^d$. The Rényi exponent is defined to be ${lim}_{\delta \to 0}\left[\right\{log{M}_{\delta }\left(h\right)\}/(-log\delta \left)\right]$, assuming that this limit exists, where $M_{\delta }\left(h\right)={\sum }_i{\mu }^h\left({\Delta }_i\right)$ and, for $\delta >0$, $\left\{{\Delta }_i\right\}$ are the cubes of a $\delta$-coordinate mesh that intersect the support of $\mu$. In particular, we demonstrate asymptotic normality of the least-squares estimator of $\tau \left(h\right)$ when the measure $\mu$ is generated by a particular class of multiplicative random cascades, a result which allows construction of interval estimates and application of hypothesis tests for this scaling exponent. Simulation results illustrating this asymptotic normality are presented.

We consider the random-design nonparametric regression model with long-range dependent errors that may also depend on the independent and identically distributed explanatory variables. Disclosing a smoothing dichotomy, we show that the finite-dimensional distributions of the Nadaraya-Watson kernel estimator of the regression function converge either to those of a degenerate process with completely dependent marginals or to those of a Gaussian white-noise process. The first case occurs when the bandwidths are large enough in a specified sense to allow long-range dependence to prevail. The second case is for bandwidths that are small in the given sense, when both the required norming sequence and the limiting process are the same as if the errors were independent. The borderline situation results in a limiting convolution of the two cases. The main results contrast with previous findings for deterministic-design regression.

We prove convergence of the p-optimal martingale measures to the minimal-entropy martingale measure for p →1. This is done for bounded stochastic processes in a discrete-time setting with a finite horizon. We also investigate in detail an example of an unbounded process, where we do not find this convergence.

Various known expressions in terms of hyperbolic functions for the Laplace transforms of random times related to one-dimensional Brownian motion are derived in a unified way by excursion theory and extended to one-dimensional diffusions.

We prove a Taylor expansion of the density $p^{\epsilon }\left(y\right)$ of a Wiener functional $F^{\epsilon }$ with Wiener-chaos decomposition $F^{\epsilon }=y+{\sum }_{n=1}^{\mathrm{\infty }}{\epsilon }_n^{\mathrm{nI}}\left(f_n\right)$, $\epsilon \in (0,1]$. Using Malliavin calculus, a precise description of the coefficients in the development in terms of the multiple integrals $I_n\left(f_n\right)$ is provided. This general result is applied to the study of the density in two examples of hyperbolic stochastic partial differential equations with linear coefficients, where the driving noise has been perturbed by a coefficient $\epsilon$.

In this paper we prove that the density $p_{t,x}\left(y\right)$ of the solution of a white-noise-driven parabolic stochastic partial differential equation (SPDE) satisfying a strong ellipticity condition is $\frac{1}{2}$ Lipschitz continuous with respect to (w.r.t.) $t$ and $1-\epsilon$ Lipschitz continuous w.r.t. $x$ for all $\epsilon \in ]0,1[$. In addition, we show that it belongs to the Besov space $B_{1,\mathrm{\infty },\mathrm{\infty }}$ w.r.t. $x$. The proof is based on the Malliavin calculus of variations and on some refined estimates for the Green kernel associated with the SPDE.

A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discrete-time observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the generators of the diffusions. Optimal estimating functions in the sense of Godambe and Heyde are found. Inference based on these is invariant under transformations of data. A result on consistency and asymptotic normality of the estimators is given for ergodic diffusions. The theory is illustrated by several examples and by a simulation study.

We study consistency and asymptotic normality of posterior distributions of the regression coefficient in a linear model when the dimension of the parameter grows with increasing sample size. Under certain growth restrictions on the dimension (depending on the design matrix), we show that the posterior distributions concentrate in neighbourhoods of the true parameter and can be approximated by an appropriate normal distribution.

We study the problem of testing a simple hypothesis for a nonparametric ''signal + white-noise'' model. It is assumed under the null hypothesis that the ''signal'' is completely specified, e.g., that no signal is present. This hypothesis is tested against a composite alternative of the following form: the underlying function (the signal) is separated away from the null in the L₂ norm and, in addition, it possesses some smoothness properties. We focus on the case of an inhomogeneous alternative when the smoothness properties of the signal are measured in an L_p norm with p<2. We consider tests whose errors have probabilities which do not exceed prescribed values and we measure the quality of testing by the minimal distance between the null and the alternative set for which such testing is still possible. We evaluate the optimal rate of decay for this distance to zero as the noise level tends to zero. Then a rate-optimal test is proposed which essentially uses a pointwise-adaptive estimation procedure.

We consider first the spline smoothing nonparametric estimation with variable smoothing parameter and arbitrary design density function and show that the corresponding equivalent kernel can be approximated by the Green function of a certain linear differential operator. Furthermore, we propose to use the standard (in applied mathematics and engineering) method for asymptotic solution of linear differential equations, known as the Wentzel-Kramers-Brillouin method, for systematic derivation of an asymptotically equivalent kernel in this general case. The corresponding results for polynomial splines are a special case of the general solution. Then, we show how these ideas can be directly extended to the very general L-spline smoothing.