Table of contents, Bernoulli Journal, vol. 10, no. 1 (2004)

Counting process intensity estimation by orthogonal wavelet methods; Authors: Prakash N. Patil, Andrew T.A. Wood

Aggregating regression procedures to improve performance; Authors: Yuhong Yang

Estimation and testing in a partial linear regression model under long-memory dependence; Authors: Germán Aneiros-Pérez, Wenceslao González-Manteiga, Philippe Vieu

The Bernoulli sieve; Authors: Alexander V. Gnedin

On multidimensional Ornstein-Uhlenbeck processes driven by a general Lévy process; Authors: Hiroki Masuda

Slow, fast and arbitrary growth conditions for renewal-reward processes when both the renewals and the rewards are heavy-tailed; Authors: Vladas Pipiras, Murad S. Taqqu, Joshua B. Levy

A stochastic wave equation in dimension 3: smoothness of the law; Authors: Lluís Quer-Sardanyons, Marta Sanz-Solé

Editorial board, Bernoulli Journal, vol. 10, no. 1 (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)

In recent years wavelet-based methods have been proposed to estimate density and regression functions, most often in settings with independent and identically distributed observations. An attractive feature of these methods is that the rate of convergence is unaffected by the presence of discontinuities in the function being estimated. We provide structure and mean-square analyses of wavelet-based estimators of counting process intensities in the context of the multiplicative intensity model.

A fundamental question regarding combining procedures concerns the potential gain and how much one needs to pay for it in terms of statistical risk. Juditsky and Nemirovski considered the case where a large number of procedures are to be combined. We give upper and lower bounds for complementary cases. Under an $l_1$ constraint on the linear coefficients, it is shown that for pursuing the best linear combination of $n^\tau$ procedures, in terms of rate of convergence under the squared $L_2$ loss, one can pay a price of order $O$(log $n/n^{1-\tau})$ when $0 < \tau < \frac{1}{2}$ and a price of order $O$((log $n/n)^{\frac {1}{2}}$ when $\frac{1}{2} \leq \tau < \infty$. These rates cannot be improved or essentially improved in a uniform sense. This result suggests that one should be cautious in pursuing the best linear combination, because one may end up paying a high price for nothing when linear combination in fact does not help. We show that with care in aggregation, the final procedure can automatically avoid paying the high price for such a case and then behaves as well as the best candidate procedure.

We discuss estimation and testing of hypotheses in a partial linear regression model, that is, a regression model where the regression function is the sum of a linear and a nonparametric component. We focus on the case where the covariables and the random noise do not necessarily have summable autocovariance functions, and the estimators and test statistics are based on kernel smoothing. We obtain the bias, variance and asymptotic distribution of both estimators for the parametric and nonparametric parts, as well as the asymptotic distributions of the statistics used, both under the null hypothesis and local alternatives. We thus generalize the results of Speckman and of Beran and Ghosh to the case of general structures for the autocovariance function and complete the results of González-Manteiga and Vilar-Fernández to the case of a partial linear regression model. Simulations and a real data example provide promising results for our tests.

The Bernoulli sieve is a recursive construction of a random composition (ordered partition) of an integer n. This composition can be induced by sampling from a random discrete distribution which has frequencies equal to the sizes of components of a stick-breaking interval partition of [0,1]. We exploit the Markov property of the composition and its renewal representation to study the number of its parts. We derive asymptotics of the moments and prove a central limit theorem.

We prove the following probabilistic properties of a multidimensional Ornstein-Uhlenbeck process driven by a general Lévy process, under mild regularity conditions: the strong Feller property; the existence of a smooth transition density; and the exponential β-mixing property. As a class of possible invariant distributions of an Ornstein-Uhlenbeck process, we also discuss centred and non-skewed multidimensional generalized hyperbolic distributions.

Consider M independent and identically distributed renewal-reward processes with heavy-tailed renewals and rewards that have either finite variance or heavy tails. Let $W^{*}(\mathrm{Ty},M),y\in [0,1]$ , denote the total reward process computed as the sum of all rewards in M renewal-reward processes over the time interval [0,T]. If T→∞ and then M→∞, Taqqu and Levy have shown that the properly normalized total reward process $W^{*}(T\cdot ,M)$ converges to the stable Lévy motion, but, if M→∞ followed by T→∞, the limit depends on whether the tails of the rewards are lighter or heavier than those of renewals. If they are lighter, then the limit is a self-similar process with stationary and dependent increments. If the rewards have finite variance, this self-similar process is fractional Brownian motion, and if they are heavy-tailed rewards, it is a stable non-Gaussian process with infinite variance. We consider asymmetric rewards and investigate what happens when M and T go to infinity jointly, that is, when M is a function of T and M=M(T)→∞ as T→∞. We provide conditions on the growth of M for the total reward process $W^{*}(T\cdot ,M(T\left)\right)$ to converge to any of the limits stated above, as T→∞. We also show that when the tails of the rewards are heavier than the tails of the renewals, the limit is stable Lévy motion as M=M(T)→∞, irrespective of the function M(T).

We prove the existence and regularity of the density of the real-valued solution to a three-dimensional stochastic wave equation. The noise is white in time and has a spatially homogeneous correlation whose spectral measure μ satisfies , for some $\eta \in (0,\frac{1}{2})$. Our approach uses the mild formulation of the equation given by means of Dalang's extended version of Walsh's stochastic integration. We apply the tools of Malliavin calculus on the appropriate Gaussian space related to the noise. An extension of Dalang's stochastic integral to the Hilbert-valued setting is needed. Let S₃ be the fundamental solution to the three-dimensional wave equation. The assumption on the noise yields upper and lower bounds for the integral ${\int }_0^{t}ds{\int }_{R^3}\mu (d\xi )\left|calF{S}_3\right(s\left)\right(\xi ){|}^2$ and upper bounds for ${\int }_0^{t}ds{\int }_{R^3}\mu (d\xi )\left|\xi \right|\left|calF{S}_3\right(s\left)\right(\xi ){|}^2$ in terms of powers of t. These estimates, together with a suitable mollifying procedure for S₃, are crucial in the analysis of the inverse of the Malliavin variance.