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

TL;DR

This paper introduces a new method for constructing goodness-of-fit tests for tail copulas in extreme value theory, which are asymptotically distribution-free and applicable in multivariate cases, with demonstrated effectiveness through simulations.

Contribution

It proposes a novel transformation-based approach for distribution-free goodness-of-fit testing of tail copulas, extending to multivariate cases and validated by simulations.

Findings

Transformations lead to tests with high power.

Good finite-sample approximations are achieved.

Method extends to multivariate 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 $\bigvee_{i=1}^nX_i$ and $\bigvee_{i=1}^nY_i$ 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.