Nonparametric estimation of the dependence function for a multivariate extreme value distribution (original) (raw)

Nonparametric Estimation of the Dependence Function in Bivariate Extreme Value Distributions

Journal of Multivariate Analysis, 2001

The paper considers the problem of estimating the dependence function of a bivariate extreme survival function with standard exponential marginals. Nonparametric estimators for the dependence function are proposed and their strong uniform convergence under suitable conditions is demonstrated. Comparisons of the proposed estimators with other estimators are made in terms of bias and mean squared error. Several real data sets from various applications are used to illustrate the procedures.

Non-parametric estimators of multivariate extreme dependence functions

Journal of Nonparametric Statistics, 2005

This article reviews various characterizations of a multivariate extreme dependence function A(·). The most important estimators derived from these characterizations are also sketched. Then, a unifying approach, which puts all these estimators under the same framework, is presented. This unifying approach enables us to construct new estimators and, most importantly, to propose an automatic selection method for an unknown parameter on which all the existing non-parametric estimators of A(·) depend. Constrained smoothing splines and convex hull techniques are used to force the obtained estimators to be extreme dependence functions. A simulation study comparing these estimators on a wide range of extreme dependence functions is provided.

A characterization of gumbel's family of extreme value distributions

Statistics & Probability Letters, 1989

In this note, a family of multivariate extremal distributions proposed by Gumbel (1960) is characterized among those whose dependence function is an Archimedean copula. The domains of attraction of Gumbel's distributions are also determined within this class.

Nonparametric estimation of multivariate extreme-value copulas

Journal of Statistical Planning and Inference, 2012

Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples. An extreme-value copula is determined by its Pickands dependence function, which is a function on the unit simplex subject to certain shape constraints that arise from an integral transform of an underlying measure called spectral measure. Multivariate extensions are provided of certain rank-based nonparametric estimators of the Pickands dependence function. The shape constraint that the estimator should itself be a Pickands dependence function is enforced by replacing an initial estimator by its best least-squares approximation in the set of Pickands dependence functions having a discrete spectral measure supported on a sufficiently fine grid. Weak convergence of the standardized estimators is demonstrated and the finite-sample performance of the estimators is investigated by means of a simulation experiment.

Estimation of extreme quantiles for functions of dependent random variables

Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2014

We propose a new method for estimating the extreme quantiles for a function of several dependent random variables. In contrast to the conventional approach based on extreme value theory, we do not impose the condition that the tail of the underlying distribution admits an approximate parametric form, and, furthermore, our estimation makes use of the full observed data. The proposed method is semiparametric as no parametric forms are assumed on all the marginal distributions. But we select appropriate bivariate copulas to model the joint dependence structure by taking the advantage of the recent development in constructing large dimensional vine copulas. Consequently a sample quantile resulted from a large bootstrap sample drawn from the fitted joint distribution is taken as the estimator for the extreme quantile. This estimator is proved to be consistent. The reliable and robust performance of the proposed method is further illustrated by simulation.

Non-Parametric Estimation of the Limit Dependence Function of Multivariate Extremes

1999

This paper presents a new estimation procedure for the limit distribution of the maximum of a multivariate random sample. This procedure relies on a new and simple relationship between the copula of the underlying multivariate distribution function and the dependence function of its maximum attractor. The obtained characterization is then used to de®ne a class of kernel-based estimates for the dependence function of the maximum attractor. The consistency and the asymptotic distribution of these estimates are considered.

Dependence in multivariate extreme values

Institute of Mathematical Statistics Lecture Notes - Monograph Series, 1990

We review the limiting behavior of extreme values of sequences of random vectors in R d by considering mainly the dependence properties of its nondegenerate limit laws. We treat separately the i.i.d. case, the stationary case, the independent non-identically distributed case, and the general nonstationary case. As dependence concepts we discuss total dependence, association, positive lower orthant dependence, and independence.

Modelling the Dependence of Parametric Bivariate Extreme Value Copulas

Asian Journal of Mathematics & Statistics, 2009

In this study, we consider the situation where contraints are made on the domains of two random variables whose joint copula is an extreme value model. We introduce a new measure which characterize these conditional dependence. We proved that every bivariate extreme value copulas is totally characterized by a conditional dependence function. Every twodimensional distribution is also shown to be max-infinite divisible under a restriction on the new measure. The average and median values of the measure have been computed for the main bivariate families of parametric extreme value copulas.

A bayesian estimator for the dependence function of a bivariate extreme-value distribution

Canadian Journal of Statistics, 2008

Any continuous bivariate distribution can be expressed in terms of its margins and a unique copula. In the case of extreme-value distributions, the copula is characterized by a dependence function while each margin depends on three parameters. The authors propose a Bayesian approach for the simultaneous estimation of the dependence function and the parameters defining the margins. They describe a nonparametric model for the dependence function and a reversible jump Markov chain Monte Carlo algorithm for the computation of the Bayesian estimator. They show through simulations that their estimator has a smaller mean integrated squared error than classical nonparametric estimators, especially in small samples. They illustrate their approach on a hydrological data set. Un estimateur bayésien de la fonction de dépendance d'une loi des valeurs extrêmes bivariée Résumé : Toute loi bivariée continue peut s'écrire en fonction de ses marges et d'une copule unique. Dans le cas des lois des valeurs extrêmes, la copule est caractérisée par une fonction de dépendance tandis que chaque marge dépend de trois paramètres. Les auteurs proposent une approche bayésienne pour l'estimation simultanée de la fonction de dépendance et des paramètres définissant les marges. Ils décrivent un modèle non paramétrique pour la fonction de dépendance et un algorithme MCMCà sauts réversibles pour le calcul de l'estimateur bayésien. Ils montrent par simulation que l'erreur quadratique moyenne intégrée de leur estimateur est plus faible que celle des estimateurs classiques, surtout dans de petitséchantillons. Ils illustrent leur proposà l'aide de données hydrologiques.