Sensitivity of the limit shape of sample clouds from meta densities. (English) Zbl 1264.60035

Balkema, Guus; Embrechts, Paul; Nolde, Natalia

Bernoulli 18, No. 4, 1386-1404 (2012).

Authors’ 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 by the authors [J. Multivariate Anal. 101, No. 7, 1738–1754 (2010; Zbl 1198.60012)]. As shown there, for the light-tailed meta distribution, the sample clouds, properly scaled, converge to 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’ theorem, the existence of the directional derivatives at the upper endpoint of the copula is necessary and sufficient for the 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 the 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 if the asymptotic behaviour of the heavy-tailed density is not affected.

Reviewer: Wiesław Dziubdziela (Kielce)

MSC:

60G70	Extreme value theory; extremal stochastic processes
60E99	Distribution theory

Keywords:

extremes; limit set; limit shape; meta distribution; regular partition; sensitivity

Citations:

Zbl 1198.60012

Software:

QRM

References:

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[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. · Zbl 1198.60012 · doi:10.1016/j.jmva.2010.02.010
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