Bernoulli

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

Polynomial Pickands functions

Simon Guillotte and François Perron

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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
Received: May 2013
Revised: June 2014
First available in Project Euclid: 30 September 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1443620848

Digital Object Identifier
doi:10.3150/14-BEJ656

Mathematical Reviews number (MathSciNet)
MR3449781

Zentralblatt MATH identifier
06543268

Keywords
Bernstein’s theorem extreme value copulas Lorentz degree Pickands dependence function polynomials spectral measure

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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