The Annals of Statistics

Asymptotically distribution-free goodness-of-fit testing for tail copulas

Sami Umut Can, John H. J. Einmahl, Estate V. Khmaladze, and Roger J. A. Laeven

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Let $(X_{1},Y_{1}),\ldots,(X_{n},Y_{n})$ be an i.i.d. sample from a bivariate distribution function that lies in the max-domain of attraction of an extreme value distribution. The asymptotic joint distribution of the standardized component-wise maxima $\bigvee_{i=1}^{n}X_{i}$ and $\bigvee_{i=1}^{n}Y_{i}$ is then characterized by the marginal extreme value indices and the tail copula $R$. We propose a procedure for constructing asymptotically distribution-free goodness-of-fit tests for the tail copula $R$. The procedure is based on a transformation of a suitable empirical process derived from a semi-parametric estimator of $R$. The transformed empirical process converges weakly to a standard Wiener process, paving the way for a multitude of asymptotically distribution-free goodness-of-fit tests. We also extend our results to the $m$-variate ($m>2$) case. In a simulation study we show that the limit theorems provide good approximations for finite samples and that tests based on the transformed empirical process have high power.

Article information

Ann. Statist., Volume 43, Number 2 (2015), 878-902.

First available in Project Euclid: 23 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 62G20: Asymptotic properties 62G32: Statistics of extreme values; tail inference
Secondary: 62F03: Hypothesis testing

Extreme value theory tail dependence goodness-of-fit testing martingale transform


Can, Sami Umut; Einmahl, John H. J.; Khmaladze, Estate V.; Laeven, Roger J. A. Asymptotically distribution-free goodness-of-fit testing for tail copulas. Ann. Statist. 43 (2015), no. 2, 878--902. doi:10.1214/14-AOS1304.

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Supplemental materials

  • Supplementary material: Supplement to “Asymptotically distribution-free goodness-of-fit testing for tail copulas”. We provide a proof of Theorem 2.1 as well as details about the Monte Carlo simulations of Section 6.