Polynomial Pickands functions

摘要

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 >= 0, and show that these can be parameterized by a vector in Rm+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 Rm+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.