Extreme-Value Copulas

Being the limits of copulas of componentwise maxima in independent random samples, extreme-value copulas can be considered to provide appropriate models for the dependence structure between rare events. Extreme-value copulas not only arise naturally in the domain of extreme-value theory, they can also be a convenient choice to model general positive dependence structures. The aim of this survey is to present the reader with the state-of-the-art in dependence modeling via extreme-value copulas. Both probabilistic and statistical issues are reviewed, in a nonparametric as well as a parametric context.Comment: 20 pages, 3 figures. Minor revision, typos corrected. To appear in F. Durante, W. Haerdle, P. Jaworski, and T. Rychlik (editors) "Workshop on Copula Theory and its Applications", Lecture Notes in Statistics -- Proceedings, Springer 201

Nonparametric estimation of multivariate extreme-value copulas

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.Comment: 26 pages; submitted; Universit\'e catholique de Louvain, Institut de statistique, biostatistique et sciences actuarielle

Nonparametric estimation of multivariate extreme-value copulas

Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples and are determined by the so-called Pickands dependence function. In this thesis, the focus lies on the nonparametric estimation of multivariate extreme-value copulas. In a first project, the hypothetical situation is considered where the marginal distributions are supposed to be known. The existing nonparametric estima- tors require the choice of weight functions, of which we succeed to compute the variance-minimizing versions together with an estimator for the Pickands depen- dence function based on ordinary least squares in a linear regression framework. The following chapters of the thesis are dedicated to the more realistic situ- ation of unknown marginal distributions which are replaced by their empirical counterparts. As the shape constraints for the Pickands dependence function are in general not satisfied by the nonparametric estimators referred to above, we will enforce the constraints by replacing the 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. The last chapter presents a new test for multivariate extreme-value depen- dence based on the Pickands representation and implemented using the multi- plier resampling method. 1G(ACTU 3) -- UCL, 201

Nonparametric estimation of an extreme-value copula in arbitrary dimensions

Inference on an extreme-value copula usually proceeds via its Pickands dependence function, which is a convex function on the unit simplex satisfying certain inequality constraints. In the setting of an i.i.d. random sample from a multivariate distribution with known margins and an unknown extreme-value copula, an extension of the Capéraà-Fougères-Genest estimator was introduced by D. Zhang, M. T. Wells and L. Peng [Nonparametric estimation of the dependence function for a multivariate extreme-value distribution, Journal of Multivariate Analysis 99 (4) (2008) 577-588]. The joint asymptotic distribution of the estimator as a random function on the simplex was not provided. Moreover, implementation of the estimator requires the choice of a number of weight functions on the simplex, the issue of their optimal selection being left unresolved. A new, simplified representation of the CFG-estimator combined with standard empirical process theory provides the means to uncover its asymptotic distribution in the space of continuous, real-valued functions on the simplex. Moreover, the ordinary least-squares estimator of the intercept in a certain linear regression model provides an adaptive version of the CFG-estimator whose asymptotic behavior is the same as if the variance-minimizing weight functions were used. As illustrated in a simulation study, the gain in efficiency can be quite sizable.Empirical process Linear regression Minimum-variance estimator Multivariate extreme-value distribution Ordinary least squares Pickands dependence function Unit simplex

Nonparametric estimation of an extreme-value copula in arbitrary dimensions

Inference on an extreme-value copula usually proceeds via its Pickands dependence function, which is a convex function on the unit simplex satisfying certain inequality constraints. In the setting of an i.i.d. random sample from a multivariate distribution with known margins and an unknown extreme-value copula, an extension of the Caperaa-Fougeres-Genest estimator was introduced by D. Zhang, M. T. Wells and L Peng [Nonparametric estimation of the dependence function for a multivariate extreme-value distribution, journal of Multivariate Analysis 99 (4) (2008) 577-588]. The joint asymptotic distribution of the estimator as a random function on the simplex was not provided. Moreover, implementation of the estimator requires the choice of a number of weight functions on the simplex, the issue of their optimal selection being left unresolved. A new, simplified representation of the CFG-estimator combined with standard empirical process theory provides the means to uncover its asymptotic distribution in the space of continuous, real-valued functions on the simplex. Moreover, the ordinary least-squares estimator of the intercept in a certain linear regression model provides an adaptive version of the CFG-estimator whose asymptotic behavior is the same as if the variance-minimizing weight functions were used. As illustrated in a simulation study, the gain in efficiency can be quite sizable. (C) 2010 Elsevier Inc. All rights reserved