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Samples with a limit shape, multivariate extremes, and risk

Part of: Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  15 July 2020

Guus Balkema  and
Natalia Nolde
Guus Balkema*
Affiliation:
University of Amsterdam
Natalia Nolde*
Affiliation:
University of British Columbia
*
*Postal address: Department of Mathematics, 1012 WX Amsterdam, The Netherlands.
**Postal address: Department of Statistics, 2207 Main Mall, Vancouver, BC V6T 1Z4, Canada. Email: natalia@stat.ubc.ca
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Abstract

Large samples from a light-tailed distribution often have a well-defined shape. This paper examines the implications of the assumption that there is a limit shape. We show that the limit shape determines the upper quantiles for a large class of random variables. These variables may be described loosely as continuous homogeneous functionals of the underlying random vector. They play an important role in evaluating risk in a multivariate setting. The paper also looks at various coefficients of tail dependence and at the distribution of the scaled sample points for large samples. The paper assumes convergence in probability rather than almost sure convergence. This results in an elegant theory. In particular, there is a simple characterization of domains of attraction.

Keywords

Exponential decayintermediate tail dependencelight-tailed distributionslimit setpositive-homogeneous functionalresidual tail dependencerisk regiontail independenceWeibull-like tail

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60E05: Distributions
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 2 , June 2020 , pp. 491 - 522
Copyright
© Applied Probability Trust 2020

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