Table of contents, Bernoulli Journal, vol. 12, no. 5 (2006)

On singular values of matrices with independent rows; Authors: Shahar Mendelson, Alain Pajor

Asymptotic distribution and local power of the log-likelihood ratio test for mixtures: bounded and unbounded cases; Authors: Jean-Marc Azaïs, Élisabeth Gassiat, Cécile Mercadier

Product-limit estimators of the survival function for two modified forms of current-status data; Authors: Valentin Patilea, Jean-Marie Rolin

Locating lines among scattered points; Authors: Peter Hall, Nader Tajvidi, P.E. Malin

Covariance chains; Authors: Nanny Wermuth, D.R. Cox, Giovanni M. Marchetti

Convergence rates of posterior distributions for Brownian semimartingale models; Authors: F.H. Van Der Meulen, Aad W. Van Der Vaart, J.H. Van Zanten

Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations; Authors: Jean-Philippe Lemor, Emmanuel Gobet, Xavier Warin

Multivariate generalized Pareto distributions; Authors: Holger Rootzén, Nader Tajvidi

Geometric growth for stochastic difference equations with application to branching populations; Authors: Miguel González, Manuel Molina, Inés Del Puerto

Editorial board, Bernoulli Journal, vol. 12, no. 5 (2006)

Peter McCullagh, Editor-in-Chief

Peter J. Bickel, Editor

D.R. Cox, Editor

David Nualart, Editor

Edward Waymire, Editor

David Applebaum, Associate Editor

Lucien Birgé, Associate Editor

Evarist Giné, Associate Editor

Peter Hall, Associate Editor

Wilfrid S. Kendall, Associate Editor

Thomas Mikosch, Associate Editor

Jesper Møller, Associate Editor

Gesine Reinert, Associate Editor

Ya'acov 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)

We present deviation inequalities of random operators of the form $N^{-1}{\sum }_{i=1}^N{X}_i\otimes X_i$ from the average operator $E(X\otimes X)$, where $X_i$ are independent random vectors distributed as $X$, which is a random vector in $R^n$ or in ${\ell }_2$. We use these inequalities to estimate the singular values of random matrices with independent rows (without assuming that the entries are independent).

We consider the log-likelihood ratio test (LRT) for testing the number of components in a mixture of populations in a parametric family. We provide the asymptotic distribution of the LRT statistic under the null hypothesis as well as under contiguous alternatives when the parameter set is bounded. Moreover, for the simple contamination model we prove, under general assumptions, that the asymptotic local power under contiguous hypotheses may be arbitrarily close to the asymptotic level when the set of parameters is large enough. In the particular problem of normal distributions, we prove that, when the unknown mean is not a priori bounded, the asymptotic local power under contiguous hypotheses is equal to the asymptotic level.

The problem of estimating the distribution of a lifetime that may be left or right censored is considered. Two data structures that extend the classical current-status data framework are introduced and the corresponding product-limit estimators are derived. The strong uniform convergence and asymptotic normality of the product-limit estimators are proved. A bootstrap procedure that can be applied to confidence intervals construction is proposed.

Consider a process of events on a line L, where, for the most part, the events occur randomly in both time and location. A scatterplot of the pair that represents position on the line, and occurrence time, will resemble a bivariate stochastic point process in a plane, P say. If, however, some of the points on L arise through a more regular phenomenon which travels along the line at an approximately constant speed, creating new points as it goes, then the corresponding points in P will occur roughly in a straight line. It is of interest to locate such lines, and thereby identify, as nearly as possible, the points on L which are associated with the (approximately) constant-velocity process. Such a problem arises in connection with the study of seismic data, where L represents a fault-line and the constant-velocity process there results from the steady diffusion of stress. We suggest methodology for solving this needle-in-a-haystack problem, and discuss its properties. The technique is applied to both simulated and real data. In the latter case it draws particular attention to events occurring along the San Andreas fault, in the vicinity of Parkville, California, on 5 April 1995.

Covariance matrices which can be arranged in tridiagonal form are called covariance chains. They are used to clarify some issues of parameter equivalence and of independence equivalence for linear models in which a set of latent variables influences a set of observed variables. For this purpose, orthogonal decompositions for covariance chains are derived first in explicit form. Covariance chains are also contrasted to concentration chains, for which estimation is explicit and simple. For this purpose, maximum-likelihood equations are derived first for exponential families when some parameters satisfy zero value constraints. From these equations explicit estimates are obtained, which are asymptotically efficient, and they are applied to covariance chains. Simulation results confirm the satisfactory behaviour of the explicit covariance chain estimates also in moderate-sized samples.

We consider the asymptotic behaviour of posterior distributions based on continuous observations from a Brownian semimartingale model. We present a general result that bounds the posterior rate of convergence in terms of the complexity of the model and the amount of prior mass given to balls centred around the true parameter. This result is illustrated for three special cases of the model: the Gaussian white-noise model, the perturbed dynamical system and the ergodic diffusion model. Some examples for specific priors are discussed as well.

This study focuses on the numerical resolution of backward stochastic differential equations with data dependent on a jump-diffusion process. We propose and analyse a numerical scheme based on iterative regression functions which are approximated by projections on vector spaces of functions, with coefficients evaluated using Monte Carlo simulations. Regarding the error, we derive explicit bounds with respect to the time step, the number of paths simulated and the number of functions: this allows us to optimally adjust the parameters to achieve a given accuracy. We also present numerical tests related to option pricing with differential interest rates and locally risk-minimizing strategies (Föllmer-Schweizer decomposition).

Statistical inference for extremes has been a subject of intensive research over the past couple of decades. One approach is based on modelling exceedances of a random variable over a high threshold with the generalized Pareto (GP) distribution. This has proved to be an important way to apply extreme value theory in practice and is widely used. We introduce a multivariate analogue of the GP distribution and show that it is characterized by each of following two properties: first, exceedances asymptotically have a multivariate GP distribution if and only if maxima asymptotically are extreme value distributed; and second, the multivariate GP distribution is the only one which is preserved under change of exceedance levels. We also discuss a bivariate example and lower-dimensional marginal distributions.

We investigate the asymptotic behaviour of discrete-time processes that satisfy a stochastic difference equation. We provide conditions to guarantee geometric growth on the whole set where these processes go to infinity. The class of processes considered includes homogeneous Markov chains. The results are of interest in population dynamics. In this work they are applied to two branching populations.