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

Efficient estimation in nonlinear autoregressive time-series models; Authors: Hira L. Koul, Anton Schick

The estimation of the order of a mixture model; Authors: Didier Dacunha-Castelle, Elisabeth Gassiat

A spectral method for estimating parameters in rainfall models; Authors: Richard E. Chandler

Spread, estimators and nuisance parameters; Authors: Edwin R. Van Den Heuvel, Chris A.J. Klaassen

Berry-Esseen bounds for statistics of weakly dependent samples; Authors: V. Bentkus, F. Götze, A. Tikhomoirov

Asymptotic normality of least-squares estimators of tail indices; Authors: Sándor Csörgö, László Viharos

Editorial board, Bernoulli Journal, vol. 3, no. 3 (1997)

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

Dmitrii O. Kramkov, Associate Editor

Peter McCullagh, Associate Editor

Alexander A. Novikov, Associate Editor

Yosihiko Ogata, Associate Editor

Victor Pérez-Abreu, Associate Editor

Dag Tjøstheim, Associate Editor

This paper discusses efficient estimation for a class of nonlinear time-series models with unknown error densities. It establishes local asymptotic normality in this semi-parametric setting. This is then used to describe efficient estimates and to discuss the question of adaptation. Stein's necessary condition for adaptive estimation is satisfied if the error densities are symmetric, but is also satisfied in some models with asymmetric error densities. The paper gives several methods of constructing efficient estimates. These results are then applied to construct efficient estimators in SETAR(2;1,1), EXPAR(1) and ARMA(1,1) models. We observe that adaptation is not possible in the SETAR(2;1,1) model with asymmetric errors while the efficient estimators in the ARMA(1,1) model are adaptive even for asymmetric error densities. Section 8 contains a result that is useful in verifying the continuity of the stationary density with respect to the underlying parameters.

We propose a new method to estimate the number of different populations when a large sample of a mixture of these populations is observed. It is possible to define the number of different populations as the number of points in the support of the mixing distribution. For discrete distributions having a finite support, the number of support points can be characterized by Hankel matrices of the first algebraic moments, or Toeplitz matrices of the trigonometric moments. Namely, for one-dimensional distributions, the cardinality of the support may be proved to be the least integer such that the Hankel matrix (or the Toeplitz matrix) degenerates. Our estimator is based on this property. We first prove the convergence of the estimator, and then its exponential convergence under wide assumptions. The number of populations is not a priori bounded. Our method applies to a large number of models such as translation mixtures with known or unknown variance, scale mixtures, exponential families and various multivariate models. The method has an obvious computational advantage since it avoids any computation of estimates of the mixing parameters. Finally we give some numerical examples to illustrate the effectiveness of the method in the most popular cases.

One of the most difficult problems in rainfall modelling is often the fitting of theoretical models to data. In this paper a spectral approach is proposed for fitting single-site models, by considering the approximate likelihood functions of collections of sample Fourier coefficients. An objective function is derived, which is the same as that used in Whittle's method for time series parameter estimation, and is shown to have an interpretation as a quasi-likelihood when the time series is non-Gaussian. The method requires knowledge of the theoretical spectral density of models to be fitted; the form of this spectral density is given for a wide class of point-process-based rainfall models. A variant of the method is also developed for when a number of independent replications of a rainfall process are available. Large-sample properties of the estimators are derived, and the method is illustrated with some data from the south-west of England.

A general spread inequality for arbitrary estimators of a one-dimensional parameter is given. This finite-sample inequality yields bounds on the distribution of estimators in the presence of finite- or infinite-dimensional nuisance parameters.

We prove Berry--Esseen bounds for a general class of asymptotically normal statistics which are functions of $N$ weakly dependent random variables under easily verifiable conditions. In particular, we show, for some $\delta >0$, the validity of the bound $O\left(N^{-1/2}{log}^{\delta }N\right)$ for $U$-statistics, studentized means, functions of sample means, functionals of empirical distribution functions and linear combinations of order statistics.

Based on least-squares considerations, Schultze and Steinebach proposed three new estimators for the tail index of a regularly varying distribution function and proved their consistency. We show that, unlike the Hill estimator, all three least-squares estimators can be centred to have normal asymptotic distributions universally over the whole model, and for two of these estimators this in fact happens at the desirable order of the norming sequence. We analyse the conditions under which asymptotic confidence intervals become possible. In a submodel, we compare the asymptotic mean squared errors of optimal versions of these and earlier estimators. The choice of the number of extreme order statistics to be used is also discussed through the investigation of the asymptotic mean squared error for a comprehensive set of examples of a general kind.