Bivariate Extreme Value Copulas Research Articles

Bayesian estimation of bivariate Pickands dependence function

In the present study, Bayesian method of estimating the Pickands dependence function of bivariate extreme-value copulas is proposed. Initially, cubic B-spline regression is used to model the dependence function. Then, the estimator of Pickands dependence function is obtained by the Bayesian approach. Through the estimation process, the prior and the posterior distributions of the parameter vectors are provided. The posterior sampling algorithm is presented in order to approximate the posterior distribution. We give a simulation study to measure and compare the performance of the proposed Bayesian estimator of the Pickands dependence function. A real data example is also illustrated.

On the size of the class of bivariate extreme-value copulas with a fixed value of Spearman's rho or Kendall's tau

Abstract A bivariate extreme-value copula is characterized by a function of one variable, called a Pickands dependence function, which is convex and comprised between two bounds. The authors identify the smallest possible compact set containing the graph of all Pickands dependence functions whose corresponding bivariate extreme-value copula has a fixed value of Spearman's rho or Kendall's tau. The consequences of this result for statistical modeling are outlined.

Tail-weighted dependence measures with limit being the tail dependence coefficient

ABSTRACTFor bivariate continuous data, measures of monotonic dependence are based on the rank transformations of the two variables. For bivariate extreme value copulas, there is a family of estimators , for , of the extremal coefficient, based on a transform of the absolute difference of the α power of the ranks. In the case of general bivariate copulas, we obtain the probability limit of as the sample size goes to infinity and show that (i) for is a measure of central dependence with properties similar to Kendall's tau and Spearman's rank correlation, (ii) is a tail-weighted dependence measure for large α, and (iii) the limit as is the upper tail dependence coefficient. We obtain asymptotic properties for the rank-based measure and estimate tail dependence coefficients through extrapolation on . A data example illustrates the use of the new dependence measures for tail inference.

Polynomial Pickands functions

Pickands dependence functions characterize bivariate extreme value copulas. In this paper, we study the class of polynomial Pickands functions. We provide a solution for the characterization of such polynomials of degree at most $m+2$, $m\geq0$, and show that these can be parameterized by a vector in $\mathbb{R}^{m+1}$ belonging to the intersection of two ellipsoids. We also study the class of Bernstein approximations of order $m+2$ of Pickands functions which are shown to be (polynomial) Pickands functions and parameterized by a vector in $\mathbb{R}^{m+1}$ belonging to a polytope. We give necessary and sufficient conditions for which a polynomial Pickands function is in fact a Bernstein approximation of some Pickands function. Approximation results of Pickands dependence functions by polynomials are given. Finally, inferential methodology is discussed and comparisons based on simulated data are provided.

The spatial structure of European wind storms as characterized by bivariate extreme-value Copulas

Abstract. The winds associated with extra-tropical cyclones are amongst the costliest natural perils in Europe. Re/insurance companies typically have insured exposure at multiple locations and hence the losses they incur from any individual storm crucially depend on that storm's spatial structure. Motivated by this, this study investigates the spatial structure of the most extreme windstorms in Europe. The data consists of a carefully constructed set of 135 of the most damaging storms in the period 1972–2010. Extreme value copulas are applied to this data to investigate the spatial dependencies of gusts. The copula method is used to investigate three aspects of windstorms. First, spatial maps of expected hazard damage between large cities and their surrounding areas are presented. Second, we demonstrate a practical application of the copula method to benchmark catalogues of artificial storms for use in the re/insurance sector. Third, the copula-based method is used to investigate the sensitivity of spatially aggregated damage to climate variability. The copula method allows changes to be expressed in terms of storm frequency, local intensity, and storm spatial structure and gives a more detailed view of how climate variability may affect multi-location risk in Europe.

Modelling the Dependence of Parametric Bivariate Extreme Value Copulas

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 two- dimensional 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.

Bivariate extreme-value copulas with discrete Pickands dependence measure

A two-parametric family of bivariate extreme-value copulas (EVCs), which corresponds to precisely the bivariate EVCs whose Pickands dependence measure is discrete with at most two atoms, is introduced and analyzed. It is shown how bivariate EVCs with arbitrary discrete Pickands dependence measure can be represented as the geometric mean of such basis copulas. General bivariate EVCs can thus be represented as the limit of this construction when the number of involved basis copulas tends to infinity. Besides the theoretical value of such a representation, it is shown how several properties of the represented copula can be deduced from properties of the involved basis copulas. An algorithm for the computation of the representation is given.

Formula omitted]-measure of non-exchangeability for bivariate extreme value and Archimax copulas

In the class of bivariate extreme value copulas, an upper bound is calculated for the measure of non-exchangeability μ ∞ based on the L ∞ -norm distance between a copula C and its transpose C t ( x , y ) = C ( y , x ) . Copulas that are maximally non-exchangeable with respect to μ ∞ are also determined. Moreover, similar upper bounds are given, respectively, for the class of all EV copulas having a fixed upper tail dependence coefficient and for the larger class of Archimax copulas.

Rank-based inference for bivariate extreme-value copulas

Consider a continuous random pair (X, Y) whose dependence is characterized by an extreme-value copula with Pickands dependence function A. When the marginal distributions of X and Y are known, several consistent estimators of A are available. Most of them are variants of the estimators due to Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859–878] and Capéraà, Fougères and Genest [Biometrika 84 (1997) 567–577]. In this paper, rank-based versions of these estimators are proposed for the more common case where the margins of X and Y are unknown. Results on the limit behavior of a class of weighted bivariate empirical processes are used to show the consistency and asymptotic normality of these rank-based estimators. Their finite- and large-sample performance is then compared to that of their known-margin analogues, as well as with endpoint-corrected versions thereof. Explicit formulas and consistent estimates for their asymptotic variances are also given.

Nonparametric estimation of the dependence function for a multivariate extreme value distribution

Understanding and modeling dependence structures for multivariate extreme values are of interest in a number of application areas. One of the well-known approaches is to investigate the Pickands dependence function. In the bivariate setting, there exist several estimators for estimating the Pickands dependence function which assume known marginal distributions [J. Pickands, Multivariate extreme value distributions, Bull. Internat. Statist. Inst., 49 (1981) 859–878; P. Deheuvels, On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions, Statist. Probab. Lett. 12 (1991) 429–439; P. Hall, N. Tajvidi, Distribution and dependence-function estimation for bivariate extreme-value distributions, Bernoulli 6 (2000) 835–844; P. Capéraà, A.-L. Fougères, C. Genest, A nonparametric estimation procedure for bivariate extreme value copulas, Biometrika 84 (1997) 567–577]. In this paper, we generalize the bivariate results to p-variate multivariate extreme value distributions with p ⩾ 2 . We demonstrate that the proposed estimators are consistent and asymptotically normal as well as have excellent small sample behavior.

Non-Parametric Inference for Bivariate Extreme-Value Copulas

Extreme-value copulas arise as the possible limits of copulas of component-wise 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.

Hutchinson-Lai's conjecture for bivariate extreme value copulas

The class of bivariate extreme value copulas, which satisfies the monotone regression positive dependence property or equivalently the stochastic increasing property, is considered. A variational calculus proof of the Hutchinson-Lai conjecture about Kendall's tau and Spearman's rho for this class is provided.