Annals of Statistics

Inference for Archimax copulas

Simon Chatelain, Anne-Laure Fougères, and Johanna G. Nešlehová

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Archimax copula models can account for any type of asymptotic dependence between extremes and at the same time capture joint risks at medium levels. An Archimax copula is characterized by two functional parameters: the stable tail dependence function $\ell $, and the Archimedean generator $\psi $ which distorts the extreme-value dependence structure. This article develops semiparametric inference for Archimax copulas: a nonparametric estimator of $\ell $ and a moment-based estimator of $\psi $ assuming the latter belongs to a parametric family. Conditions under which $\psi $ and $\ell $ are identifiable are derived. The asymptotic behavior of the estimators is then established under broad regularity conditions; performance in small samples is assessed through a comprehensive simulation study. The Archimax copula model with the Clayton generator is then used to analyze monthly rainfall maxima at three stations in French Brittany. The model is seen to fit the data very well, both in the lower and in the upper tail. The nonparametric estimator of $\ell $ reveals asymmetric extremal dependence between the stations, which reflects heavy precipitation patterns in the area. Technical proofs, simulation results and $\mathsf{R}$ code are provided in the Online Supplement.

Article information

Ann. Statist., Volume 48, Number 2 (2020), 1025-1051.

Received: June 2018
Revised: February 2019
First available in Project Euclid: 26 May 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 62H12: Estimation 62G05: Estimation 62G20: Asymptotic properties 62G32: Statistics of extreme values; tail inference
Secondary: 60G70: Extreme value theory; extremal processes

Copulas empirical processes multivariate extremes subasymptotic modeling


Chatelain, Simon; Fougères, Anne-Laure; Nešlehová, Johanna G. Inference for Archimax copulas. Ann. Statist. 48 (2020), no. 2, 1025--1051. doi:10.1214/19-AOS1836.

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Supplemental materials

  • Supplement to “Inference for Archimax copulas”. This file contains the proofs of Sections 2, 4 and 6. It also contains detailed results of the simulation study from Section 5.
  • R code for “Inference for Archimax copulas”. The functions necessary for fitting the Clayton-Archimax model as was done in Section 8 are provided.