Polynomial Pickands functions

TL;DR

This paper characterizes polynomial Pickands dependence functions, providing parameterizations, approximation methods, and inferential techniques for modeling bivariate extreme value copulas.

Contribution

It offers a complete characterization of polynomial Pickands functions, including their parameterization, approximation by Bernstein polynomials, and inferential methods.

Findings

Polynomial Pickands functions are characterized by vectors in intersecting ellipsoids.

Bernstein approximations of Pickands functions are parameterized by vectors in a polytope.

Simulation studies compare different approximation and inference methods.

Abstract

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.