Table of contents, Bernoulli Journal, vol. 11, no. 2 (2005)

Diffusion-type models with given marginal distribution and autocorrelation function; Authors: Bo Martin Bibby, Ib Michael Skovgaard, Michael Sørensen

Wavelet estimation of a multifractal function; Authors: Fabrice Gamboa, Jean-Michel Loubes

Large-noise asymptotics for one-dimensional diffusions; Authors: Szymon Peszat, Francesco Russo

Cramér-Rao revisited; Authors: Andries Lenstra

Filtered Brownian motions as weak limit of filtered Poisson processes; Authors: Laurent Decreusefond, Nicholas Savy

Nonparametric bootstrap prediction; Authors: Tadayoshi Fushiki, Fumiyasu Komaki, Kazuyuki Aihara

Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model; Authors: Cristina Butucea, Catherine Matias

Adaptive estimation of linear functionals under different performance measures; Authors: T. Tony Cai, Mark G. Low

On covariance estimation of non-synchronously observed diffusion processes; Authors: Takaki Hayashi, Nakahiro Yoshida

Editorial board, Bernoulli Journal, vol. 11, no. 2 (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

Xiao-Li Meng, Associate Editor

Thomas Mikosch, Associate Editor

Dominique Picard, Associate Editor

Gesine Reinert, Associate Editor

Yaacov 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)

Flexible stationary diffusion-type models are developed that can fit both the marginal distribution and the correlation structure found in many time series from, for example, finance and turbulence. Diffusion models with linear drift and a known and prespecified marginal distribution are studied, and the diffusion coefficients corresponding to a large number of common probability distributions are found explicitly. An approximation to the diffusion coefficient based on saddlepoint techniques is developed for use in cases where there is no explicit expression for the diffusion coefficient. It is demonstrated theoretically as well as in a study of turbulence data that sums of diffusions with linear drift can fit complex correlation structures. Any infinitely divisible distribution satisfying a weak regularity condition can be obtained as a marginal distribution.

We prove that multifractal functions, characterized by their wavelet representation, can be estimated in the white noise model by a Bayesian method. We give rates of convergence for two different models. Further, we study empirical methods for estimating the hyperparameters of the model, which lead to a fully tractable estimator.

We establish a law of large numbers and a central limit theorem for a class of additive functionals related to the solution of a one-dimensional stochastic differential equation perturbed by a large noise.

In a right-angled triangle, the hypotenuse is the longest side. So, if all (hypotenuse) vectors from a given set of vectors have the same orthogonal projection onto a certain subspace, we have a lower bound for their lengths. Interpreting the square of such a length as the variance of an unbiased estimator produces an information bound. The Cramér-Rao bound and the van Trees inequality can be seen as consequences of this bound. Another consequence is an inequality for the minimax variance, that is, the maximal variance in shrinking neighbourhoods, minimized over all unbiased estimators. This bound is non-asymptotic and requires almost no regularity conditions.

The main result of this paper is a limit theorem which shows the convergence in law, on a Hölderian space, of filtered Poisson processes (a class of processes which contains shot noise process) to filtered Brownian motion (a class of processes which contains fractional Brownian motion) when the intensity of the underlying Poisson process is increasing. We apply the theory of convergence of Hilbert space valued semimartingales and use a radonification result.

Ensemble learning has recently been intensively studied in the field of machine learning. `Bagging' is a method of ensemble learning and uses bootstrap data to construct various predictors. The required prediction is then obtained by averaging the predictors. Harris proposed using this technique with the parametric bootstrap predictive distribution to construct predictive distributions, and showed that the parametric bootstrap predictive distribution gives asymptotically better prediction than a plug-in distribution with the maximum likelihood estimator. In this paper, we investigate nonparametric bootstrap predictive distributions. The nonparametric bootstrap predictive distribution is precisely that obtained by applying bagging to the statistical prediction problem. We show that the nonparametric bootstrap predictive distribution gives predictions asymptotically as good as the parametric bootstrap predictive distribution.

We consider a semiparametric convolution model where the noise has known Fourier transform which decays asymptotically as an exponential with unknown scale parameter; the deconvolution density is less smooth than the noise in the sense that the tails of the Fourier transform decay more slowly, ensuring the identifiability of the model. We construct a consistent estimation procedure for the noise level and prove that its rate is optimal in the minimax sense. Two convergence rates are distinguished according to different smoothness properties for the unknown density. If the tail of its Fourier transform does not decay faster than exponentially, the asymptotic optimal rate and exact constant are evaluated, while if it does not decay faster than polynomially, this rate is evaluated up to a constant. Moreover, we construct a consistent estimator of the unknown density, by using a plug-in method in the classical kernel estimation procedure. We establish that the rates of estimation of the deconvolution density are slower than in the case of an entirely known noise distribution. In fact, nonparametric rates of convergence are equal to the rate of estimation of the noise level, and we prove that these rates are minimax. In a few specific cases the plug-in method converges at even slower rates.

Lower bounds are given for probabilistic error subject to a mean squared error constraint. Consequences for the expected length of variable length confidence intervals centred on adaptive estimators are given. It is shown that in many contexts centring confidence intervals on adaptive estimators must lead either to poor coverage probability or unnecessarily long intervals.

We consider the problem of estimating the covariance of two diffusion processes when they are observed only at discrete times in a non-synchronous manner. The modern, popular approach in the literature, the realized covariance estimator, which is based on (regularly spaced) synchronous data, is problematic because the choice of regular interval size and data interpolation scheme may lead to unreliable estimation. We propose a new estimator which is free of any `synchronization' processing of the original data, hence free of bias or other problems caused by it.