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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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References

Balkema, A. and Embrechts, P. (2007). High Risk Scenarios and Extremes: A Geometric Approach. European Mathematical Society, Zurich.CrossRefGoogle Scholar
Balkema, A., Embrechts, P. and Nolde, N. (2010). Meta densities and the shape of their sample clouds. J. Multivariate Anal. 101, 1738–1754.CrossRefGoogle Scholar
Balkema, A. and Nolde, N. (2010). Asymptotic independence for unimodal densities. Adv. Appl. Prob. 42, 411–432.CrossRefGoogle Scholar
Balkema, A. and Nolde, N. (2012). Asymptotic dependence for homothetic light-tailed densities. Adv. Appl. Prob. 44, 506–527.CrossRefGoogle Scholar
Barndorff-Nielsen, O. (1963). On the limit behavior of extreme order statistics. Ann. Math. Statist. 34, 992–1002.CrossRefGoogle Scholar
Bingham, N., Goldie, C. and Teugels, J. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Cline, D. (1988). Joint stable attraction of two sums of products. J. Multivariate Anal. 25, 272–285.CrossRefGoogle Scholar
Coles, S. and Tawn, J. (1994). Statistical methods for multivariate extremes: an application to structural design (with discussion). Appl. Statist. 43, 1–48.CrossRefGoogle Scholar
Davis, R., Mulrow, E. and Resnick, S. (1988). Almost sure limit sets of random samples in ${\mathbb R}^d$. Adv. Appl. Prob. 20, 573–599.CrossRefGoogle Scholar
De Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer-Verlag, New York.CrossRefGoogle Scholar
De Haan, L. and Zhou, C. (2011). Extreme residual dependence for random vectors and processes. Adv. Appl. Prob. 43, 217–242.CrossRefGoogle Scholar
De Valk, C. (2016). Approximation and estimation of very small probabilities of multivariate extreme events. Extremes 19, 687–717.CrossRefGoogle Scholar
De Valk, C. (2016). Approximation of high quantiles from intermediate quantiles. Extremes 19, 661–686.CrossRefGoogle Scholar
Dekkers, A. and de Haan, L. (1993). Optimal choice of sample fraction in extreme value estimation. J. Multivariate Anal. 47, 173–195.CrossRefGoogle Scholar
Eddy, W. (1980). The distribution of the convex hull of a Gaussian sample. J. Appl. Prob. 17, 686–695.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin, Heidelberg.CrossRefGoogle Scholar
Fisher, L. (1969). Limiting sets and convex hulls of samples from product measures. Ann. Math. Statist. 40, 1824–1832.CrossRefGoogle Scholar
Frick, M. (2009). Multivariate extremal density expansions and residual tail dependence structures. Master’s Thesis, University of Siegen.Google Scholar
Geffroy, J. (1958). Contribution à la theorie des valeurs extrêmes. Publ. Inst. Statist. Univ. Paris VII, 37–121.Google Scholar
Geffroy, J. (1959). Contribution à la theorie des valeurs extrêmes. Publ. Inst. Statist. Univ. Paris VIII, 3–65.Google Scholar
Gnedenko, B. (1943). Sur la distribution limite du terme maximum d’une série aléatoire. Ann. Math. 44, 423–453.CrossRefGoogle Scholar
Hashorva, E. (2010). On the residual dependence index of elliptical distributions. Statist. Prob. Lett. 80, 1070–1078.CrossRefGoogle Scholar
Joe, H. (1993). Parametric families of multivariate distributions with given margins. J. Multivariate Anal. 46, 262–282.CrossRefGoogle Scholar
Kinoshita, K. and Resnick, S. (1991). Convergence of scaled random samples in ${\mathbb R}^d$. Ann. Prob. 19, 1640–1663.CrossRefGoogle Scholar
Ledford, A. and Tawn, J. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187.CrossRefGoogle Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley & Sons, New York.Google Scholar
Nolde, N. (2014). Geometric interpretation of the residual dependence coefficient. J. Multivariate Anal. 123, 85–95.CrossRefGoogle Scholar
Sibuya, M. (1960). Bivariate extreme statistics. Ann. Inst. Statist. Math. 11, 195–210.CrossRefGoogle Scholar