Table of contents, Bernoulli Journal, vol. 3, no. 2 (1997)

Sieve bootstrap for time series; Authors: Peter Bühlmann

Second-order properties of an extrapolated bootstrap without replacement under weak assumptions; Authors: Patrice Bertail

Estimation of integral functionals of a density and its derivatives; Authors: Béatrice Laurent

Tauber theory for infinitely divisible variance functions; Authors: Bent Jørgensen, José Raúl Martínez

A Poisson approximation for the Dirichlet law, the Ewens sampling formula and the Griffiths-Engen-McCloskey law by the Stein-Chen coupling method; Authors: Ulrich Martin Hirth

Stochastic differential equations with random coefficients; Authors: Arturo Kohatsu-Higa, Jorge A. León, David Nualart

We study a bootstrap method which is based on the method of sieves. A linear process is approximated by a sequence of autoregressive processes of order p=p(n), where p(n)→∞, p(n)=o(n) as the sample size n→∞. For given data, we then estimate such an AR(p(n)) model and generate a bootstrap sample by resampling from the residuals. This sieve bootstrap enjoys a nice nonparametric property, being model-free within a class of linear processes.

This paper shows that a straightforward extrapolation of the bootstrap distribution obtained by resampling without replacement, as considered by Politis and Romano, leads to second-order correct confidence intervals, provided that the resampling size is chosen adequately. We assume only that the statistic of interest T_n, suitably renormalized by a regular sequence, is asymptotically pivotal and admits an Edgeworth expansion on some differentiable functions. The results are extended to a corrected version of the moving-block bootstrap without replacement introduced by Künsch for strong-mixing random fields. Moreover, we show that the generalized jackknife or the Richardson extrapolation of such bootstrap distributions, as considered by Bickel and Yahav, leads to better approximations.

We consider the problem of estimating a functional of a density of the type $\int \phi (f,f^{\prime },...,f^{\left(k\right)},.)$. The estimation of $\int \phi (f,.)$ has already been studied by the author: starting from efficient estimators of linear and quadratic functionals of $f$ and its derivatives and using a Taylor expansion of $\phi$, we construct estimators which achieve the $n^{-1/2}$ rate whenever $f$ is smooth enough. Moreover, we show that these estimators are efficient. We also obtain the optimal rate of convergence when the $n^{-1/2}$ rate is not achievable and when $k>0$. Concerning the estimation of quadratic functionals, more precisely of integrated squared density derivatives, Bickel and Ritov have already constructed efficient estimators. Here we propose an alternative construction based on projections, an approach which seems more natural.

We study a notion of Tauber theory for infinitely divisible natural exponential families, showing that the variance function of the family is (bounded) regularly varying if and only if the canonical measure of the Lévy-Khinchine representation of the family is (bounded) regularly varying. Here a variance function V is called bounded regularly varying if V(μ)\sim cμ^p either at zero or infinity, with a similar definition for measures. The main tool of the proof is classical Tauber theory.

We consider the random number of (Griffiths-Engen-McCloskey (GEM))-(Poisson-Dirichlet) components which are greater than ε. In two alternative and similar ways, letting Dirichlet laws and Ewens sampling formula laws respectively converge to the GEM-(Poisson-Dirichlet) law and using the Stein-Chen coupling method, we prove the Poisson approximation with respect to the total variation metric of the satisfactory order of magnitude 1/expectation.

In this paper we establish the existence and uniqueness of a solution for different types of stochastic differential equation with random initial conditions and random coefficients. The stochastic integral is interpreted as a generalized Stratonovich integral, and the techniques used to derive these results are mainly based on the path properties of the Brownian motion, and the definition of the Stratonovich integral.