The Annals of Statistics

Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition

John H. J. Einmahl, Laurens de Haan, and Deyuan Li

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Consider n i.i.d. random vectors on ℝ2, with unknown, common distribution function F. Under a sharpening of the extreme value condition on F, we derive a weighted approximation of the corresponding tail copula process. Then we construct a test to check whether the extreme value condition holds by comparing two estimators of the limiting extreme value distribution, one obtained from the tail copula process and the other obtained by first estimating the spectral measure which is then used as a building block for the limiting extreme value distribution. We derive the limiting distribution of the test statistic from the aforementioned weighted approximation. This limiting distribution contains unknown functional parameters. Therefore, we show that a version with estimated parameters converges weakly to the true limiting distribution. Based on this result, the finite sample properties of our testing procedure are investigated through a simulation study. A real data application is also presented.

Article information

Ann. Statist., Volume 34, Number 4 (2006), 1987-2014.

First available in Project Euclid: 3 November 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G32: Statistics of extreme values; tail inference 62G30: Order statistics; empirical distribution functions 62G10: Hypothesis testing
Secondary: 60G70: Extreme value theory; extremal processes 60F17: Functional limit theorems; invariance principles

Dependence structure goodness-of-fit test bivariate extreme value theory tail copula process weighted approximation


Einmahl, John H. J.; de Haan, Laurens; Li, Deyuan. Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition. Ann. Statist. 34 (2006), no. 4, 1987--2014. doi:10.1214/009053606000000434.

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