## Bernoulli

• Bernoulli
• Volume 22, Number 1 (2016), 213-241.

### Polynomial Pickands functions

#### 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.

#### Article information

Source
Bernoulli, Volume 22, Number 1 (2016), 213-241.

Dates
Revised: June 2014
First available in Project Euclid: 30 September 2015

https://projecteuclid.org/euclid.bj/1443620848

Digital Object Identifier
doi:10.3150/14-BEJ656

Mathematical Reviews number (MathSciNet)
MR3449781

Zentralblatt MATH identifier
06543268

#### Citation

Guillotte, Simon; Perron, François. Polynomial Pickands functions. Bernoulli 22 (2016), no. 1, 213--241. doi:10.3150/14-BEJ656. https://projecteuclid.org/euclid.bj/1443620848

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