Table of contents, Bernoulli Journal, vol. 8, no. 5 (2002)

Nonparametric estimation of conditional quantiles using quantile regression trees; Authors: Probal Chaudhuri, Wei-Yin Loh

Non-asymptotic minimax rates of testing in signal detection; Authors: Yannick Baraud

Density and hazard estimation in censored regression models; Authors: Ingrid Van Keilegom, Noël Veraverbeke

Irregular sets and central limit theorems; Authors: Gonzalo Perera

Optimality and small ͉-optimality of martingale estimating functions; Authors: Martin Jacobsen

Optimal series representation of fractional Brownian sheets; Authors: Thomas Kühn, Werner Linde

Editorial board, Bernoulli Journal, vol. 8, no. 5 (2002)

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)

A nonparametric regression method that blends key features of piecewise polynomial quantile regression and tree-structured regression based on adaptive recursive partitioning of the covariate space is investigated. Unlike least-squares regression trees, which concentrate on modelling the relationship between the response and the covariates at the centre of the response distribution, our quantile regression trees can provide insight into the nature of that relationship at the centre as well as the tails of the response distribution. Our nonparametric regression quantiles have piecewise polynomial forms, where each piece is obtained by fitting a polynomial quantile regression model to the data in a terminal node of a binary decision tree. The decision tree is constructed by recursively partitioning the data based on repeated analyses of the residuals obtained after model fitting with quantile regression. One advantage of the tree structure is that it provides a simple summary of the interactions among the covariates. The asymptotic behaviour of piecewise polynomial quantile regression estimates and the associated derivative estimates are studied under appropriate regularity conditions. The methodology is illustrated with an example on the incidence rates of mumps in the United States.

Let $Y=(Y_i)_{i \in I}$ be a finite or countable sequence of independent Gaussian random variables with mean $f=(f_i)_{i \in I}$ and common variance $\sigma^2$. For various sets $\mathcal{F} \subset \ell_2(I)$, the aim of this paper is to describe the minimal $\ell_2$-distance between $f$ and $0$ for the problem of testing $f=0$ against $f\neq 0$, $f\in \mathcal {F}$, to be possible with prescribed error probabilities. To do so, we start with the set $\mathcal {F}$ which collects the sequences $f$ such that $f_j=0$ for $j>n$ and $ | \{j, f_j \neq 0\}| \leq k$, where the numbers k and n are integers satisfying $1\leq k \leq n$. Then we show how such a result allows us to handle the cases where $\mathcal {F}$ is an ellipsoid and more generally an $\ell_p$-body with $p \in ]0,2]$. Our results are not asymptotic in the sense that we do not assume that $\sigma$ tends to $0$. Finally, we consider the problem of adaptive testing.

Let (X,Y) be a random vector, where Y denotes the variable of interest, possibly subject to random right censoring, and X is a covariate. Consider a heteroscedastic model Y=m(X)+σ(X)ε, where the error term ε is independent of X and m(X) and σ(X) are smooth but unknown functions. Under this model, we construct a nonparametric estimator for the density and hazard function of Y given X, which has a faster rate of convergence than the completely nonparametric estimator that is constructed without making any model assumption. Moreover, the proposed estimator for the density and hazard function performs better than the classical nonparametric estimator, especially in the right tail of the distribution.

In previous papers we have studied the asymptotic behaviour of$S_N(A;X)=(2N+1)^{-d/2}\Sigma_{n \in A_N}X_n$, where $X$ is a centred, stationary and weakly dependent random field, and $A_N=A \cap [ -N,N]^d, A \subset \mathbb {Z}^d$. This leads to the definition of asymptotically measurable sets, which enjoy the property that $S_N(A;X)$ has a (Gaussian) weak limit for any $X$ belonging to a certain class. We present here an application of this technique. Consider a regression model $X_n=\varphi (\xi_,n,Y_n), n \in \mathbb {Z}^d$, where $X_n$ is centred, $\varphi$ satisfies certain regularity conditions, and $\xi$ and $Y$ are independent random fields; for any $m \in \mathbb {N}$ and $(y1,\ldots,y_m)$, the central limit theorem holds for ($\varphi (\xi ,y_1),\ldots, \xi ,y_m)$), but $Y$ satisfies only the strong law of large numbers as it applies to $(Y_m,Y_{m-n})_{m \in \mathbb{Z}^d}$, for any $n \in \mathbb{Z}^d$. Under these conditions, it is shown that the central limit theorem holds for $X$.

Martingale estimating functions determined from a given collection (the base) of conditional expectations are considered for estimating the parameters of a discretely observed diffusion. Small ͉-optimality of these functions (i.e. near-efficiency when the observations are close together) is discussed, and in particular it is shown that this can be achieved provided the base is large enough. It is also shown that the optimal martingale estimating function with a given base is automatically small ͉-optimal, provided only that the base is sufficiently large. In both cases the critical dimension of the base is the same and determined by the dimension of the diffusion, and by whether the squared diffusion matrix is parameter-dependent or not; the critical number does not depend, however, on the dimension of the parameter.

For 0 $lt; \gamma 2$, let $B_\gamma^d$ be a $d$-dimensional $\gamma$-fractional Brownian sheet with index set $[0,1]^d$ and let $(\xi_k)k \geq 1$ be an independent sequence of standard normal random variables. We prove the existence of continuous functions uk such that almost surely

$$B_γ^d(t)=\sum\limits_{k=1}^\infty \xi_k u_k(t), \qquad t \in[0,1]^d,$$

and

$$ \left({\mathbb E}\sup_{t\in[0,1]^d}\left|\sum\limits_{k=n}^\infty\xi_k\,u_k(t)\right|^2\right)^{1/2} \approx n^{-\gamma/2}\,(1+log n)^{d(\gamma+1)/2\,-\gamma/2} \;$$

This order is shown to be optimal. We obtain small-ball estimates for $B^\gamma_d$, extending former results in the case $\gamma=1$. Our investigations rest upon basic properties of different kinds of $s$-numbers of operators.$