Table of contents, Bernoulli Journal, vol. 8, no. 2 (2002)

Asymptotic equivalence for a null recurrent diffusion; Authors: Sylvain Delattre, Marc Hoffmann

On positive spectral density functions; Authors: Richard C. Bradley

Entrance from 0+ for increasing semi-stable Markov processes; Authors: Jean Bertoin, Maria-Emilia Caballero

The uniqueness class of continuous local martingales; Authors: Samia Beghdadi-Sakrani

Minimax or maxisets?; Authors: Gérard Kerkyacharian, Dominique Picard

Estimation of the innovation quantile density function of an AR(p) process based on autoregression quantiles; Authors: Faouzi El Bantli, Marc Hallin

Editorial board, Bernoulli Journal, vol. 8, no. 2 (2002)

Willem R. van Zwet, Editor-in-Chief

Sara A. van de Geer, Co-Editor

Peter J. Bickel, Editor

D.R. Cox, Editor

David Nualart, Editor

Edward C. Waymire, Editor

Adrian Baddeley, Associate Editor

Lucien Birgé, Associate Editor

Phillipe Carmona, Associate Editor

Paul Embrechts, Associate Editor

Evarist Giné, Associate Editor

Friedrich Götze, Associate Editor

Peter Guttorp, Associate Editor

Frank den Hollander, Associate Editor

Iain Johnstone, Associate Editor

Wilfrid S. Kendall, Associate Editor

M.R. Leadbetter, Associate Editor

Enno Mammen, Associate Editor

Peter McCullagh, Associate Editor

Thomas Mikosch, Associate Editor

Dominique Picard, Associate Editor

David Siegmund, Associate Editor

Simon Tavaré, Associate Editor

Dag Tjøstheim, Associate Editor

Tineke H. Bakker, Editorial Assistant

Richard Leigh, Copy Editor

Ank Lepping, Desk Editor

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

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

We establish that the model generated by the observation of the path of a one-dimensional null recurrent diffusion, when the parameter is the compactly supported drift, is asymptotically equivalent to a mixed Gaussian white noise experiment as the observation time T → ∞. The approximation is given in the sense of Le Cam's deficiency ͉-distance over Sobolev balls of smoothness order β > ½.

A necessary and sufficient condition is given for a weakly stationary random field (indexed by the integer lattice of an arbitrary finite dimension) to have a spectral density which is bounded between two positive constants. As a corollary, a necessary and sufficient condition is derived for a positive continuous spectral density. The conditions involve `linear' dependence coefficients.

We consider increasing semi-stable Markov processes starting at x>0 and specify their asymptotic behaviour in law as x→0+. This can be viewed as an extension of a result of Brennan and Durrett on the asymptotic size of a particle undergoing a certain type of random splitting.

We study some properties of the class \mathcal{U} of laws of a continuous local martingale which is determined by the law of its quadratic variation, and we give a `simple' characterization of this class.

We discuss a new way of evaluating the performance of a statistical estimation procedure. This consists of investigating the maximal set where a given procedure has a given rate of convergence. Although the setting is not vastly different from the minimax context, it is in a sense less pessimistic and provides a functional set which is authentically connected to the procedure and the model. We also investigate more traditional concerns about procedures: oracle inequalities. Difficulties arise in the practical definition of this notion when the loss function is not the L₂ norm. We explain these difficulties and suggest a new definition in the cases of L_p norms and pointwise estimation. We investigate the connections between maxisets and local oracle inequalities, and prove that verifying a local oracle inequality implies that the maxiset automatically contains a prescribed set linked with the oracle inequality. We have investigated the consequences of this statement on well-known efficient adaptive methods: wavelet thresholding and local bandwidth selection. We prove local oracle inequalities for these methods and draw conclusions about the maxisets associated with them.

In this paper, we propose two types of estimator (one of histogram type, the other a kernel estimate) of the quantile density (or sparsity) function α\mapsto [f(F^-1(α ))]^-1 associated with the innovation density f of an autoregressive model of order p. Our estimators are based on autoregression quantiles. Contrary to more classical estimators based on estimated residuals, they are autoregression-invariant and scale-equivariant. Their asymptotic behaviour is derived from a uniform Bahadur representation for autoregression quantiles - a result of independent interest. Simulations are carried out to illustrate their performance.