Asymptotically Distribution-Free Goodness-of-Fit Testing for Tail Copulas

S.U. Can, J.H.J. Einmahl, E.V. Khmaladze, R.J.A. Laeven

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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.
Original languageEnglish
Place of PublicationTilburg
PublisherEconometrics
Number of pages28
Volume2014-041
Publication statusPublished - 30 Jun 2014

Publication series

NameCentER Discussion Paper
Volume2014-041

Fingerprint

Distribution-free Test
Empirical Process
Distribution-free
Goodness of Fit Test
Copula
Goodness of fit
Tail
Extreme Value Index
Extreme Value Distribution
Testing
Bivariate Distribution
Domain of Attraction
Wiener Process
Limit Theorems
Joint Distribution
High Power
Asymptotic distribution
Distribution Function
Simulation Study
Converge

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.
Can, S.U. ; Einmahl, J.H.J. ; Khmaladze, E.V. ; Laeven, R.J.A. / Asymptotically Distribution-Free Goodness-of-Fit Testing for Tail Copulas. Tilburg : Econometrics, 2014. (CentER Discussion Paper).
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Can, SU, Einmahl, JHJ, Khmaladze, EV & Laeven, RJA 2014 'Asymptotically Distribution-Free Goodness-of-Fit Testing for Tail Copulas' CentER Discussion Paper, vol. 2014-041, Econometrics, Tilburg.

Asymptotically Distribution-Free Goodness-of-Fit Testing for Tail Copulas. / Can, S.U.; Einmahl, J.H.J.; Khmaladze, E.V.; Laeven, R.J.A.

Tilburg : Econometrics, 2014. (CentER Discussion Paper; Vol. 2014-041).

Research output: Working paperDiscussion paperOther research output

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AU - Khmaladze, E.V.

AU - Laeven, R.J.A.

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N2 - Let (X1, Y1),…., (Xn, Yn) be an i.i.d. sample from a bivariate distribution function that lies in the max-domain of attraction ofan 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 empiricalprocess 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.

AB - Let (X1, Y1),…., (Xn, Yn) be an i.i.d. sample from a bivariate distribution function that lies in the max-domain of attraction ofan 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 empiricalprocess 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.

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Can SU, Einmahl JHJ, Khmaladze EV, Laeven RJA. Asymptotically Distribution-Free Goodness-of-Fit Testing for Tail Copulas. Tilburg: Econometrics. 2014 Jun 30. (CentER Discussion Paper).