Table of contents, Bernoulli Journal, vol. 2, no. 1 (1996)

Empirical processes and applications: an overview; Authors: Evarist Giné

Discussion; Authors: Jon A. Wellner

Rejoinder; Authors: Evarist Giné

Neo-classical minimax problems, thresholding and adaptive function estimation; Authors: David L. Donoho, Iain M. Johnstone

Distribution of the particles of a critical branching Wiener process; Authors: Pál Révész

The variance-optimal martingale measure for continuous processes; Authors: Freddy Delbaen, Walter Schachermayer

Editorial board, Bernoulli Journal, vol. 2, no. 1 (1996)

Ole E. Barndorff-Nielsen, Editor-in-Chief

Jens Ledet Jensen, Managing Co-Editor

Peter J. Bickel, Editor

David Nualart, Editor

David R. Cox, Editor

Edward C. Waymire, Editor

Adrian Baddeley, Associate Editor

Lucien Birgé, Associate Editor

Louis H. Y. Chen, Associate Editor

Hans Föllmer, Associate Editor

Richard D. Gill, Associate Editor

Peter Guttorp, Associate Editor

Dmitrii O. Kramkov, Associate Editor

Peter McCullagh, Associate Editor

Alexander A. Novikov, Associate Editor

Yosihiko Ogata, Associate Editor

Victor Pérez-Abreu, Associate Editor

Dag Tjøstheim, Associate Editor

Some recent theory of empirical processes indexed by general classes of sets of functions is reviewed. Several of the main results, as well as some of the methods, such as randomization and reduction to Gaussian processes, are described. Applications in asymptotic statistics are illustrated by a few examples. The bootstrap of empirical processes, which enhances the applicability of the theory (as invariance of the limit law is the exception rather than the rule), is also discussed.

This is a discussion of the Forum Lectures by Evarist Ginéon the subject of Empirical Processes and Applications presented at the European Meeting of Statisticians held in Bath, England, September 13-18, 1992. The discussion includes short sketches of developments in probability theory (Gaussian process theory, weak convergence theory, and probability in Banach spaces), empirical process theory, and applications thereof in statistics. I comment briefly on the formulation of central limit theorems for empirical processes in terms of the presence or absence of a Gaussian hypothesis, expand on Professor Giné's discussion of the bootstrap, and briefly explain my recent results for exchangeably weighted bootstraps obtained jointly with Jens Praestgaard. The discussion closes with some problems.

This is a rejoinder of the Forum Lectures by Evarist Ginéon the subject of Empirical Processes and Applications presented at the European Meeting of Statisticians held in Bath, England, September 13-18, 1992. The discussion includes short sketches of developments in probability theory (Gaussian process theory, weak convergence theory, and probability in Banach spaces), empirical process theory, and applications thereof in statistics. I comment briefly on the formulation of central limit theorems for empirical processes in terms of the presence or absence of a Gaussian hypothesis, expand on Professor Giné's discussion of the bootstrap, and briefly explain my recent results for exchangeably weighted bootstraps obtained jointly with Jens Praestgaard. The discussion closes with some problems.

We study the problem of estimating θ from data Y ~ N(θ,σ²) under squared-error loss. We define three new scalar minimax problems in which the risk is weighted by the size of θ. Simple thresholding gives asymptotically minimax estimates in all three problems. We indicate the relationships of the new problems to each other and to two other neo-classical problems: the problems of the bounded normal mean and of the risk-constrained normal mean. Via the wavelet transform, these results have implications for adaptive function estimation in two settings: estimating functions of unknown type and degree of smoothness in a global ℓ² norm; and estimating a function of unknown degree of local Hölder smoothness at a fixed point. In the latter setting, the scalar minimax results imply: Lepskii's results that it is not possible fully to adapt the unknown degree of smoothness without incurring a performance cost; and that simple thresholding of the empirical wavelet transform gives an estimate of a function at a fixed point which is, to within constants, optimally adaptive to unknown degree of smoothness.

Consider a critical branching Wiener process in R^d. Let {F_T(x),T=1,2,...,x∈R^d} be the distribution of the particles living at time T. The main result of this paper tells us that any given absolutely continuous function F(x) will be well approximated by F_T(x) with positive probability if T is big enough and the process does not die out up to T.

We prove that for continuous stochastic processes S based on (Ω,F, P) for which there is an equivalent martingale measure Q⁰ with square-integrable density dQ⁰/dP, we have that the so-called `variance-optimal' martingale measure Q^opt for which the density dQ^opt/dP has minimal L²(P)-norm is automatically equivalent to P. The result is then applied to an approximation problem arising in mathematical finance.

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