### Abstract

Let (X1, Y1),…., (Xn, Yn) 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

√n i=1 Xi and √n i=1 Yi 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.

an extreme value distribution. The asymptotic joint distribution of the standardized component-wise maxima

√n i=1 Xi and √n i=1 Yi 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.

Original language | English |
---|---|

Place of Publication | Tilburg |

Publisher | Econometrics |

Number of pages | 28 |

Volume | 2014-041 |

Publication status | Published - 30 Jun 2014 |

### Publication series

Name | CentER Discussion Paper |
---|---|

Volume | 2014-041 |

### Fingerprint

### Keywords

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

### Cite this

Can, S. U., Einmahl, J. H. J., Khmaladze, E. V., & Laeven, R. J. A. (2014).

*Asymptotically Distribution-Free Goodness-of-Fit Testing for Tail Copulas*. (CentER Discussion Paper; Vol. 2014-041). Tilburg: Econometrics.