Asymptotics of empirical copula processes under non-restrictive smoothness assumptions

Weak convergence of the empirical copula process is shown to hold under the assumption that the first-order partial derivatives of the copula exist and are continuous on certain subsets of the unit hypercube. The assumption is non-restrictive in the sense that it is needed anyway to ensure that the candidate limiting process exists and has continuous trajectories. In addition, resampling methods based on the multiplier central limit theorem, which require consistent estimation of the first-order derivatives, continue to be valid. Under certain growth conditions on the second-order partial derivatives that allow for explosive behavior near the boundaries, the almost sure rate in Stute's representation of the empirical copula process can be recovered. The conditions are verified, for instance, in the case of the Gaussian copula with full-rank correlation matrix, many Archimedean copulas, and many extreme-value copulas.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ387 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Hybrid Copula Estimators

An extension of the empirical copula is considered by combining an estimator of a multivariate cumulative distribution function with estimators of the marginal cumulative distribution functions for marginal estimators that are not necessarily equal to the margins of the joint estimator. Such a hybrid estimator may be reasonable when there is additional information available for some margins in the form of additional data or stronger modelling assumptions. A functional central limit theorem is established and some examples are developed.Comment: 17 page

Max-stable models for multivariate extremes

Multivariate extreme-value analysis is concerned with the extremes in a multivariate random sample, that is, points of which at least some components have exceptionally large values. Mathematical theory suggests the use of max-stable models for univariate and multivariate extremes. A comprehensive account is given of the various ways in which max-stable models are described. Furthermore, a construction device is proposed for generating parametric families of max-stable distributions. Although the device is not new, its role as a model generator seems not yet to have been fully exploited.Comment: Invited paper for RevStat Statistical Journal. 22 pages, 3 figure

The motivic zeta function and its smallest poles

Let f be a regular function on a nonsingular complex algebraic variety of dimension d. We prove a formula for the motivic zeta function of f in terms of an embedded resolution. This formula is over the Grothendieck ring itself, and specializes to the formula of Denef and Loeser over a certain localization. We also show that the space of n-jets satisfying f=0 can be partitioned into locally closed subsets which are isomorphic to a cartesian product of some variety with an affine space of dimension the round up of dn/2. Finally, we look at the consequences for the poles of the motivic zeta function

Rare Events, Temporal Dependence and the Extremal Index

AMS classifications: 60G70; 62G32;block maximum;exceedance;extremal index;failure set;mixing condition;M4 process;rare event;stationary sequence

Approximate Distributions of Clusters of Extremes

In a stationary sequence of random variables, high-threshold exceedances may cluster together.Two approximations of such a clusters distribution are established.These justify and generalize sampling schemes for clusters of extremes already known for Markov chains.approximation theory;sampling;markov chains

Non-Parametric Inference for Bivariate Extreme-Value Copulas

Extreme-value copulas arise as the possible limits of copulas of componentwise maxima of independent, identically distributed samples.The use of bivariate extreme-value copulas is greatly facilitated by their representation in terms of Pickands dependence functions.The two main families of estimators of this dependence function are (variants of) the Pickands estimator and the Caperaa-Fougeres-Genest estimator.In this paper, a unified treatment is given of these two families of estimators, and within these classes those estimators with the minimal asymptotic variance are determined.Main result is the explicit construction of an adaptive, minimum-variance estimator within a class of estimators that encompasses the Caperaa-Fougeres-Genest estimator.estimator;nonparametric inference

Tails of correlation mixtures of elliptical copulas

Correlation mixtures of elliptical copulas arise when the correlation parameter is driven itself by a latent random process. For such copulas, both penultimate and asymptotic tail dependence are much larger than for ordinary elliptical copulas with the same unconditional correlation. Furthermore, for Gaussian and Student t-copulas, tail dependence at sub-asymptotic levels is generally larger than in the limit, which can have serious consequences for estimation and evaluation of extreme risk. Finally, although correlation mixtures of Gaussian copulas inherit the property of asymptotic independence, at the same time they fall in the newly defined category of near asymptotic dependence. The consequences of these findings for modeling are assessed by means of a simulation study and a case study involving financial time series.Comment: 21 pages, 3 figure