Table of contents, Bernoulli Journal, vol. 10, no. 4 (2004)

Nonparametric independent component analysis; Authors: Alexander Samarov, Alexandre Tsybakov

Semiparametric density estimation under a two-sample density ratio model; Authors: K.F. Cheng, C.K. Chu

Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes; Authors: Christian Francq, Jean-Michel Zakoïan

On the global geometry of parametric models and information recovery; Authors: Paul Marriott, Paul Vos

Extending Doob's consistency theorem to nonparametric densities; Authors: Antonio Lijoi, Igor Prünster, Stephen G. Walker

On local estimates and the Stein method; Authors: Vydas Cekanavicius

The Matsumoto-Yor property on trees; Authors: Hélène Massam, Jacek Wesołowski

Random scale perturbation of an AR(1) process and its properties as a nonlinear explicit filter; Authors: Valentine Genon-Catalot, Mathieu Kessler

The law of the iterated logarithm for the integrated squared deviation of a kernel density estimator; Authors: Evarist Giné, David M. Mason

Editorial board, Bernoulli Journal, vol. 10, no. 4 (2004)

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)

We consider the problem of nonparametric estimation of a d-dimensional probability density and its `principal directions' in the independent component analysis model. A new method of estimation based on diagonalization of nonparametric estimates of certain matrix functionals of the density is suggested. We show that the proposed estimators of principal directions are $\sqrt{n}$-consistent and that the corresponding density estimators converge at the optimal rate.

A semiparametric density estimation is proposed under a two-sample density ratio model. This model, arising naturally from case-control studies and logistic discriminant analyses, can also be regarded as a biased sampling model. Our proposed density estimate is therefore an extension of the kernel density estimate suggested by Jones for length-biased data. We show that under the model considered the new density estimator not only is consistent but also has the `smallest' asymptotic variance among general nonparametric density estimators. We also show how to use the new estimate to define a procedure for testing the goodness of fit of the density ratio model. Such a test is consistent under very general alternatives. Finally, we present some results from simulations and from the analysis of two real data sets.

We prove the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator of the parameters of pure generalized autoregressive conditional heteroscedastic (GARCH) processes, and of autoregressive moving-average models with noise sequence driven by a GARCH model. Results are obtained under mild conditions.

We examine the question of which statistic or statistics should be used in order to recover information important for inference. We take a global geometric viewpoint, developing the local geometry of Amari. By examining the behaviour of simple geometric models, we show how not only the local curvature properties of parametric families but also the global geometric structure can be of crucial importance in finite-sample analysis. The tool we use to explore this global geometry is the Karhunen-Loève decomposition. Using global geometry, we show that the maximum likelihood estimate is the most important one-dimensional summary of information, but that traditional methods of information recovery beyond the maximum likelihood estimate can perform poorly. We also use the global geometry to construct better information summaries to be used with the maximum likelihood estimate.

We extend Doob's well-known result on Bayesian consistency. The extension covers the case where the nonparametric prior is fully supported by densities. However, our use of martingales differs from that of Doob. We also consider rates.

We show that recursive application of the Stein equation reduces uniform and non-uniform local estimates to estimates of the difference operator in total variation. We illustrate our approach by an example of an sth-order signed compound Poisson approximation to the sum of integer-valued random variables.

Viewing the Matsumoto-Yor property as a bivariate property with respect to the simple tree with two vertices and one edge, we extend it to a p-variate property with respect to any tree with p vertices. The converse of the Matsumoto-Yor property, which characterizes the product of a gamma and a generalized inverse Gaussian distribution, is extended to characterize the product of a gamma and p-1 generalized inverse Gaussian distributions. A striking feature of this characterization is that we need the independence of the components of random vectors corresponding only to the leaves of the tree. We illustrate our results with two particular trees: the two-link chain and the three-branch 'daisy'.

We study the properties of a nonlinear model of filtering in discrete time which leads to explicit computations. The signal is a standard AR(1) process, but noises are multiplicative and non-Gaussian. If the initial distribution of the AR(1) process is taken to belong to a specified class, the prediction and optimal filters also belong to this class and the prediction and updating steps are explicit. We prove the existence of a stationary version for the prediction filter and complete the theoretical study with simulations to illustrate the behaviour of the filter.

Let $f_{n,K}$ denote a kernel estimator of a density f in $R$ such that ${\int }_R{f}^p\left(x\right)dx<\mathrm{\infty }$ for some p>2. It is shown, under quite general conditions on the kernel K and on the window sizes, that the centred integrated squared deviation of $f_{n,K}$ from its mean, $\parallel f_{n,K}-E{f}_{n,K}{\parallel }_2^2-E\parallel f_{n,K}-E{f}_{n,K}{\parallel }_2^2$ satisfies a law of the iterated logarithm (LIL). This is then used to obtain an LIL for the deviation from the true density, $\parallel f_{n,K}-f{\parallel }_2^2-E\parallel f_{n,K}-f{\parallel }_2^2$. The main tools are the Komlós-Major-Tusnády approximation, a moderate-deviation result for triangular arrays of weighted chi-square variables adapted from work by Pinsky, and an exponential inequality due to Giné, Latała and Zinn for degenerate U-statistics applied in combination with decoupling and maximal inequalities.