Anne-Laure Fougères

Models for Dependent Extremes Using Stable Mixtures

This paper unifies and extends results on a class of multivariate extreme value (EV) models studied by Hougaard, Crowder and Tawn. In these models, both unconditional and conditional distributions are themselves EV distributions, and all lower-dimensional marginals and maxima belong to the class. One interpretation of the models is as size mixtures of EV distributions, where the mixing is by positive stable distributions. A second interpretation is as exponential-stable location mixtures (for Gumbel) or as power-stable scale mixtures (for non-Gumbel EV distributions). A third interpretation is through a peaks over thresholds model with a positive stable intensity. The mixing variables are used as a modelling tool and for better understanding and model checking. We study EV analogues of components of variance models, and new time series, spatial and continuous parameter models for extreme values. The results are applied to data from a pitting corrosion investigation. Copyright (c) 2008 Board of the Foundation of the Scandinavian Journal of Statistics.

A Stochastic Ordering Based on a Decomposition of Kendall’s Tau

If X and Y are random variables with joint distribution function H, their dependence may be measured by Kendall’s tau, τ(X,Y), expressed as 4E(V) — 1 in terms of the random quantity V = H(X,Y) with distribution K on the interval [0,1]. A new dependence ordering based on K is defined and studied; although it does not always imply the classical positive quadrant dependence ordering, it is shown to be weaker than the association ordering of (1987) under weak regularity conditions.

Estimation of bivariate excess probabilities for elliptical models

Let

Multivariate Archimax copulas

(X,Y)

Limit conditional distributions for bivariate vectors with polar representation

be a random vector whose conditional excess probability

Estimation of conditional laws given an extreme component

\theta(x,y) := P(Y \leq y ~ | \; X >x)

Tailweight with respect to the mode for unimodal distributions

is of interest. Estimating this kind of probability is a delicate problem as soon as

Bias correction in multivariate extremes

x

Dense classes of multivariate extreme value distributions

tends to be large, since the conditioning event becomes an extreme set. Assume that

Pitting corrosion: Comparison of treatments with extreme value distributed responses.

(X,Y)