Table of contents, Bernoulli Journal, vol. 4, no. 2 (1998)

On a universal strong law of large numbers for conditional expectations; Authors: Andrzej S. Kozek, Julian R. Leslie, Eugene F. Schuster

Correction of earthquake location estimation in a small-seismic-array system; Authors: Yosihiko Ogata, Akio Kobayashi, Naoya Mikami, Yasuaki Murata, Koichi Katsura

Discretely observing a white noise change-point model in the presence of blur; Authors: Jacques Istas, Henrik Stryhn

On large-deviation efficiency in statistical inference; Authors: Anatolii Puhalskii, Vladimir Spokoiny

On global and pointwise adaptive estimation; Authors: Sam Efromovich

Editorial board, Bernoulli Journal, vol. 4, no. 2 (1998)

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

Peter McCullagh, Associate Editor

Alexander A. Novikov, Associate Editor

Yosihiko Ogata, Associate Editor

Victor Pérez-Abreu, Associate Editor

Terence P. Speed, Associate Editor

Dag Tjøstheim, Associate Editor

Ank Lepping, Desk Editor

A number of generalizations of the Kolmogorov strong law of large numbers are known including convex combinations of random variables (rvs) with random coefficients. In the case of pairs of i.i.d. rvs $(X_1,Y_1),...,(X_n,Y_n)$, with $\mu$ being the probability distribution of the $x$s, the averages of the $Y$s for which the accompanying $X$s are in a vicinity of a given point $x$ may converge with probability 1 (w.p. 1) and for $\mu$-almost everywhere ($\mu$ a.e.) $x$ to conditional expectation $r\left(x\right)=E\left(Y\right|X=x)$. We consider the Nadaraya-Watson estimator of $E\left(Y\right|X=x)$ where the vicinities of $x$ are determined by window widths $h_n$. Its convergence towards $r\left(x\right)$ w.p. 1 and for $\mu$ a.e. $x$ under the condition $E\left|Y\right|<\mathrm{\infty }$ is called a strong law of large numbers for conditional expectations (SLLNCE). If no other assumptions on $\mu$ except that implied by $E\left|Y\right|<\mathrm{\infty }$ are required then the SLLNCE is called universal. In the present paper we investigate the minimal assumptions for the SLLNCE and for the universal SLLNCE. We improve the best-known results in this direction.

An array of regularly spaced seismic stations can estimate the location of a distant earthquake using arrival times at the stations of seismic waves generated by the earthquakes. However, the accuracy decreases as the distance to the epicentre of the earthquake from the array increases. This paper is concerned with the modification of the estimated location by removing its bias which is locally systematic but globally complex, reflecting the structure of the Earth's interior. Spline surfaces are used to model such biases. Then a Bayesian procedure is carried out not only to tune the smoothness constraints but also to select the best combination among various sums of squares of differently weighted residuals and various roughness penalties for the smoothing. Using the estimated splines of the posterior mode, the newly determined epicentre locations are transformed to confirm its practical utility. Residual distributions show that our procedure improves the modification by the conventional procedure. A spatial pattern of the residuals reveals some geophysical characteristics.

In discretely observed diffusion models, inference about unknown parameters in a smooth drift function has attracted much interest of late. This paper deals with a diffusion-type change-point model where the drift has a discontinuity across the point of change, analysed in detail in continuous time by Ibragimov and Hasminskii. We consider discrete versions of this model with integrated or blurred observations at a regular lattice. Asymptotic convergence rates and limiting distributions are given for the maximum likelihood change-point estimator when the observation noise and the lattice spacing simultaneously decrease. In particular, it is shown that the continuous and discrete model convergence rates are generally equal only up to a constant; under specific conditions on the blurring function this constant equals unity, and the normalized difference between the estimators tends to zero in the limit.

We present a general approach to statistical problems with criteria based on probabilities of large deviations. Our main idea, which originates from similarity in the definitions of the large-deviation principle (LDP) and weak convergence, is to develop a large-deviation analogue of asymptotic decision theory. We introduce the concept of the LPD for sequences of statistical experiments, which parallels the concept of weak convergence of experiments, and prove that, in analogy with Le Cam's minimax theorem, the LPD provides an asymptotic lower bound for the sequence of appropriately defined minimax risks. We also show that the bound is tight and give a method of constructing decisions whose asymptotic risk is arbitrarily close to the bound. The construction is further specified for hypothesis testing and estimation problems.

Let an estimated function belong to a Lipschitz class of order $\alpha$. Consider a minimax approach where the infimum is taken over all possible estimators and the supremum is taken over the considered class of estimated functions. It is known that, if the order $\alpha$ is unknown, then the minimax mean squared (pointwise) error convergence slows down from $n^{-2\alpha /(2\alpha +1)}$ for the case of the given $\alpha$ to $[n/ln(n){]}^{-2\alpha /(2\alpha +1)}$. At the same time, the minimax mean integrated squared (global) error convergence is proportional to $n^{-2\alpha /(2\alpha +1)}$ for the cases of known and unknown $\alpha$. We show that a similar phenomenon holds for analytic functions where the lack of knowledge of the maximal set to which the functioncan be analytically continued leads to the loss of a sharp constant. Surprisingly, for the more general adaptive minimax setting where we consider the union of a range of Lipschitz and a range of analytic functions neither pointwise error convergence nor global error convergence suffers an additional slowing down.