## The Annals of Statistics

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

#### Abstract

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

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

Dates
First available in Project Euclid: 23 March 2015

https://projecteuclid.org/euclid.aos/1427115290

Digital Object Identifier
doi:10.1214/14-AOS1304

Mathematical Reviews number (MathSciNet)
MR3325713

Zentralblatt MATH identifier
1312.62072

#### Citation

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. https://projecteuclid.org/euclid.aos/1427115290

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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.