Bernoulli

  • Bernoulli
  • Volume 18, Number 4 (2012), 1386-1404.

Sensitivity of the limit shape of sample clouds from meta densities

Guus Balkema, Paul Embrechts, and Natalia Nolde

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Abstract

The paper focuses on a class of light-tailed multivariate probability distributions. These are obtained via a transformation of the margins from a heavy-tailed original distribution. This class was introduced in Balkema et al. (J. Multivariate Anal. 101 (2010) 1738–1754). As shown there, for the light-tailed meta distribution the sample clouds, properly scaled, converge onto a deterministic set. The shape of the limit set gives a good description of the relation between extreme observations in different directions. This paper investigates how sensitive the limit shape is to changes in the underlying heavy-tailed distribution. Copulas fit in well with multivariate extremes. By Galambos’s theorem, existence of directional derivatives in the upper endpoint of the copula is necessary and sufficient for convergence of the multivariate extremes provided the marginal maxima converge. The copula of the max-stable limit distribution does not depend on the margins. So margins seem to play a subsidiary role in multivariate extremes. The theory and examples presented in this paper cast a different light on the significance of margins. For light-tailed meta distributions, the asymptotic behaviour is very sensitive to perturbations of the underlying heavy-tailed original distribution, it may change drastically even when the asymptotic behaviour of the heavy-tailed density is not affected.

Article information

Source
Bernoulli, Volume 18, Number 4 (2012), 1386-1404.

Dates
First available in Project Euclid: 12 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1352727816

Digital Object Identifier
doi:10.3150/11-BEJ370

Mathematical Reviews number (MathSciNet)
MR2995801

Zentralblatt MATH identifier
1264.60035

Keywords
extremes limit set limit shape meta distribution regular partition sensitivity

Citation

Balkema, Guus; Embrechts, Paul; Nolde, Natalia. Sensitivity of the limit shape of sample clouds from meta densities. Bernoulli 18 (2012), no. 4, 1386--1404. doi:10.3150/11-BEJ370. https://projecteuclid.org/euclid.bj/1352727816


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References

  • [1] Balkema, A.A., Embrechts, P. and Nolde, N. (2012). The shape of asymptotic dependence. In Prokhorov and Contemporary Probability Theory (A. Shiryaev, S. Varadhan and E. Presman, eds.). Springer. To appear.
  • [2] Balkema, A.A., Embrechts, P. and Nolde, N. (2010). Meta densities and the shape of their sample clouds. J. Multivariate Anal. 101 1738–1754.
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