Table of contents, Bernoulli Journal, vol. 11, no. 4 (2005)

Minimax expected measure confidence sets for restricted location parameters; Authors: Steven N. Evans, Ben B. Hansen, Philip B. Stark

Edgeworth-type expansions for transition densities of Markov chains converging to diffusions; Authors: Valentin Konakov, Enno Mammen

On the rate of convergence of the maximum likelihood estimator in Brownian semimartingale models; Authors: Harry Van Zanten

New large-deviation local theorems for sums of independent and identically distributed random vectors when the limit distribution is α-stable; Authors: Alexander Nagaev, Alexander Zaigraev

Probabilistic approximation and inviscid limits for one-dimensional fractional conservation laws; Authors: Benjamin Jourdain, Sylvie Méléard, Wojbor A. Woyczynski

Empirical likelihood based inference for the derivative of the nonparametric regression function; Authors: Gengsheng Qin, Min Tsao

Convergence results for conditional expectations; Authors: Irene Crimaldi, Luca Pratelli

Bootstrap prediction and Bayesian prediction under misspecified models; Authors: Tadayoshi Fushiki

Editorial board, Bernoulli Journal, vol. 11, no. 4 (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

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)

We study confidence sets for a parameter θ∈Θ that have minimax expected measure among random sets with at least 1-α coverage probability. We characterize the minimax sets using duality, which helps to find confidence sets with small expected measure and to bound improvements in expected measure compared with standard confidence sets. We construct explicit minimax expected length confidence sets for a variety of one-dimensional statistical models, including the bounded normal mean with known and with unknown variance. For the bounded normal mean with unit variance, the minimax expected measure 95% confidence interval has a simple form for Θ= [-τ, τ] with τ≤3.25. For Θ= [-3, 3], the maximum expected length of the minimax interval is about 14% less than that of the minimax fixed-length affine confidence interval and about 16% less than that of the truncated conventional interval [X -1.96, X + 1.96] ∩[-3,3].

We consider triangular arrays of Markov chains that converge weakly to a diffusion process. We prove Edgeworth-type expansions of order o(n^-1-δ),δ>0, for transition densities. For this purpose we apply the paramatrix method to represent the transition density as a functional of densities of sums of independent and identically distributed variables. Then we apply Edgeworth expansions to the densities. The resulting series gives our Edgeworth-type expansion for the Markov chain transition density.

In this paper we present a unified approach to obtaining rates of convergence for the maximum likelihood estimator (MLE) in Brownian semimartingale models of the form $$d{X}_t={\beta }_t^{n,\theta }dt+{\sigma }_t^{n}d{W}_t,t\le T_n.$$ We show that the rate of the MLE is determined by (an appropriate version of) the entropy of the parameter space with respect to the random metric h_n, defined by $$h_n^2(\theta ,\psi )={\int }_0^{T_n}{\left(\frac{{\beta }_s^{n,\theta }-{\beta }_s^{n,\psi }}{{\sigma }_s^n}\right)}^{2}ds.$$ Several known results for the rates in certain popular sub-models of the Brownian semimartingale model are shown to be special cases in our general framework.

A class of absolutely continuous distributions in R^d is considered. Each distribution belongs to the domain of normal attraction of an α-stable law. The limit law is characterized by a spectral measure which is absolutely continuous with respect to the spherical Lebesgue measure. The large-deviation problem for sums of independent and identically distributed random vectors when the underlying distribution belongs to that class is studied. At the focus of attention are the deviations in the directions, where the spectral density equals zero. The main conclusion is that the deviation in such a direction is explained by two abnormally large summands.

We are interested in the one-dimensional scalar conservation law $${\partial }_{t}u(t,x)=\nu D^{\alpha }u(t,x)-{\partial }_{\mathrm{xA}}\left(u\right(t,x\left)\right)$$ with fractional viscosity operator D^αv(x) = F^-1(|ξ|^αF(v)(ξ))(x) is the cumulative distribution function of a signed measure on R. We associate a nonlinear martingale problem with the Fokker-Planck equation obtained by spatial differentiation of the conservation law. After checking uniqueness for both the conservation law and the martingale problem, we prove existence thanks to a propagation-of-chaos result for systems of interacting particles with fixed intensity of jumps related to ν. The empirical cumulative distribution functions of the particles converge to the solution of the conservation law. As a consequence, it is possible to approximate this solution numerically by simulating the stochastic differential equation which gives the evolution of particles. Finally, when the intensity of jumps vanishes (ν→0) as the number of particles tends to +∞, we obtain that the empirical cumulative distribution functions converge to the unique entropy solution of the inviscid (ν=0) conservation law.

We study statistical inference for the derivative of the nonparametric regression function using local linear model based empirical likelihood. We first derive a normal equation for the derivative through the local linear model and use this equation to construct an empirical likelihood for the derivative. We show that the limiting distribution of the empirical likelihood ratio is a scaled ${\chi }_1^2$ distribution rather than the usual (unscaled) ${\chi }_1^2$ distribution. We use this limiting distribution to construct pointwise confidence intervals for the derivative. Such empirical likelihood ratio confidence intervals are easier to obtain than the normal approximation based confidence intervals. A small simulation study also suggests that they are more accurate.

Let E,F be two Polish spaces and [X_n,Y_n],[X,Y] random variables with values in E×F (not necessarily defined on the same probability space). We show some conditions which are sufficient in order to assure that, for each bounded continuous function f on E×F, the conditional expectation of f(X_n,Y_n) given Y_n converges in distribution to the conditional expectation of f(X,Y) given Y.

We consider a statistical prediction problem under misspecified models. In a sense, Bayesian prediction is an optimal prediction method when an assumed model is true. Bootstrap prediction is obtained by applying Breiman's `bagging' method to a plug-in prediction. Bootstrap prediction can be considered to be an approximation to the Bayesian prediction under the assumption that the model is true. However, in applications, there are frequently deviations from the assumed model. In this paper, both prediction methods are compared by using the Kullback-Leibler loss under the assumption that the model does not contain the true distribution. We show that bootstrap prediction is asymptotically more effective than Bayesian prediction under misspecified models.