Table of contents, Bernoulli Journal, vol. 10, no. 3 (2004)

Laslett's line segment problem; Authors: Erik W. Van Zwet

Asymptotics for the Tukey depth process, with an application to a multivariate trimmed mean; Authors: Jean-Claude Massé

Likelihood functions based on parameter-dependent functions; Authors: Thomas A. Severini

Asymptotic normality of urn models for clinical trials with delayed response; Authors: Feifang Hu, Li-Xin Zhang

Saddlepoint approximations to the trimmed mean; Authors: Roelof Helmers, Bing-Yi Jing, Gengsheng Qin, Wang Zhou

Rate of convergence in probability to the Marchenko-Pastur law; Authors: Friedrich Götze, Alexander Tikhomirov

Stability of the tail Markov chain and the evaluation of improper priors for an exponential rate parameter; Authors: James P. Hobert, Dobrin Marchev, Jason Schweinsberg

Editorial board, Bernoulli Journal, vol. 10, no. 3 (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)

Our problem is to estimate the length distribution of fractures in a rock surface from a geological map. We do not fully observe the fractures because part of the rock surface is covered by vegetation, soil and water. The uncovered region is very irregular and, as a result, we tend to observe several pieces of a single fracture. It is quite impossible to decide from the map if two pieces belong to the same underlying fracture. Under the assumption that the observed pieces are independent, we derive the nonparametric maximum likelihood estimator of the length distribution of the underlying fractures. The assumption is clearly false, but our approach is justified by proving consistency of the estimator without appealing to the independence. We apply our estimator to the geological data.

We describe the asymptotic behaviour of the empirical Tukey depth process. It is seen that the latter may not converge weakly, even though its marginals always do. Closed subsets of the index set where weak convergence does occur are identified and a necessary and a sufficient condition for the asymptotic normality of the marginals is given. As an application, asymptotic normality of a Tukey depth-based multivariate trimmed mean is obtained for smooth distributions.

Consider likelihood inference about a scalar function ψ of a parameter θ. Two methods of constructing a likelihood function for ψ are conditioning and marginalizing. If, in the model with ψ held fixed, T is ancillary, then a marginal likelihood may be based on the distribution of T, which depends only on ψ; alternatively, if a statistic S is sufficient when ψ is fixed, then a conditional likelihood function may be based on the conditional distribution of the data given S. The statistics T and S are generally required to be the same for each value of ψ. In this paper, we consider the case in which either T or S is allowed to depend on ψ. Hence, we might consider the marginal likelihood function based on a function T_ψ or the conditional likelihood given a function S_ψ. The properties and construction of marginal and conditional likelihood functions based on parameter-dependent functions are studied. In particular, the case in which T_ψ and S_ψ may be taken to be functions of the maximum likelihood estimators is considered and approximations to the resulting likelihood functions are presented. The results are illustrated on several examples.

For response-adaptive clinical trials using a generalized Friedman's urn design, we derive the asymptotic normality of the sample fraction assigned to each treatment under staggered entry and delayed response. The rate of convergence and the law of the iterated logarithm are obtained for both the urn composition and the sample fraction. Some applications are also discussed.

Saddlepoint approximations for the trimmed mean and the studentized trimmed mean are established. Some numerical evidence on the quality of our saddlepoint approximations is also included.

It is shown that the Kolmogorov distance between the spectral distribution function of a random covariance matrix (1/p)XX^T, where X is an n×p matrix with independent entries and the distribution function of the Marchenko-Pastur law is of order O(n^-1/2) in probability. The bound is explicit and requires that the twelfth moment of the entries of the matrix is uniformly bounded and that p/n is separated from 1.

Let Z be a continuous random variable with a lower semicontinuous density f that is positive on (0,∞) and 0 elsewhere. Put G(x) = ∨_x^∞ f(z)dz. We study the tail Markov chain generated by Z, defined as the Markov chain Ψ=(Ψ_n)_n=0^∞ with state space [0, ∞) and Markov transition density k(y|x) = f(y+x)/G(x). This chain is irreducible, aperiodic and reversible with respect to G. It follows that Ψ is positive recurrent if and only if Z has a finite expectation. We prove (under regularity conditions) that if E Z = ∞, then Ψ is null recurrent if and only if ∨₁^∞ 1/[ z³ f(z) ] dz = ∞. Furthermore, we describe an interesting decision-theoretic application of this result. Specifically, suppose that X is an Exp(θ) random variable; that is, X has density θe^{- θx} for x>0. Let ν be an improper prior density for θ that is positive on (0,∞). Assume that ∨₀^∞ θ ν(θ) dθ< ∞, which implies that the posterior density induced by ν is proper. Let m_ν denote the marginal density of X induced by ν; that is, m_ν(x) = ∨₀^∞ θe^-θx ν(θ) dθ. We use our results, together with those of Eaton and of Hobert and Robert, to prove that ν is a \cal P-admissible prior if ∨₁^∞ 1/ [x² m_ν(x)]dx = ∞.