Table of contents, Bernoulli Journal, vol. 2, no. 4 (1996)

Excess functions and estimation of the extreme-value index; Authors: Jan Beirlant, Petra Vynckier, Josef L. Teugels

Prediction and asymptotics; Authors: O.E. Barndorff-Nielsen, David R. Cox

Exponential convergence of Langevin distributions and their discrete approximations; Authors: Gareth O. Roberts, Richard L. Tweedie

Classes of mixing stable processes; Authors: Jan Rosinski, Gennady Samorodnitsky

A general class of estimators of the extreme-value index is generated using estimates of mean, median and trimmed excess functions. Special cases yield earlier proposals in the literature, such as Pickands' (1975) estimator. A particular restatement of the mean excess function yields an estimator which can be derived from the slope at the right upper tail from a generalized quantile plot. From this viewpoint algorithms can be constructed to search for the number of extremes needed to minimize the mean square error of the estimator. Basic asymptotic properties of this estimator are derived. The method is applied in case studies of size distributions for alluvial diamonds and of wind speeds.

Prediction of an unobserved random variable is considered from a frequentist viewpoint. After a brief review of previous work, a number of examples in which an exact solution is possible are given, partly for their intrinsic interest and partly to illustrate general results. A new form of predictive density is derived accurate to the third order of asymptotic theory under ordinary repeated sampling. The formula is invariant under transformation of the observed and unobserved random variables and under reparametrization. It respects the conditionality principle and may be based on the minimal prediction sufficient statistic. Some open problems are noted.

In this paper we consider a continuous-time method of approximating a given distribution $\pi$ using the Langevin diffusion $d{L}_t=d{W}_t+\frac{1}{2}\nabla log\pi \left(L_t\right)dt$. We find conditions under which this diffusion converges exponentially quickly to $\pi$ or does not: in one dimension, these are essentially that for distributions with exponential tails of the form $\pi \left(x\right)\propto exp(-\gamma |x{|}^{\beta })$, $0<\beta <\mathrm{\infty }$, exponential convergence occurs if and only if $\beta \ge 1$. We then consider conditions under which the discrete approximations to the diffusion converge. We first show that even when the diffusion itself converges, naive discretizations need not do so. We then consider a 'Metropolis-adjusted' version of the algorithm, and find conditions under which this also converges at an exponential rate: perhaps surprisingly, even the Metropolized version need not converge exponentially fast even if the diffusion does. We briefly discuss a truncated form of the algorithm which, in practice, should avoid the difficulties of the other forms.

Every measurable stationary α-stable process with 0<α<2 can be related to a non-singular flow on a σ-finite measure space. We establish the relationship between properties of the flow and mixing of the stationary stable process. We provide the first example of a mixing stationary stable process corresponding to a conservative flow. We show further the connection between the expected return time of the flow to sets of finite positive measure and the mixing properties of the process.