The Shape of Asymptotic Dependence
Multivariate risk analysis is concerned with extreme observations. If the underlying distribution has a unimodal density then both the decay rate of the tails and the asymptotic shape of the level sets of the density are of importance for the dependence s
- PDF / 2,971,675 Bytes
- 25 Pages / 439.36 x 666.15 pts Page_size
- 53 Downloads / 186 Views
Abstract Multivariate risk analysis is concerned with extreme observations. If the underlying distribution has a unimodal density then both the decay rate of the tails and the asymptotic shape of the level sets of the density are of importance for the dependence structure of extreme observations. For heavy-tailed densities, the sample clouds converge in distribution to a Poisson point process with a homogeneous intensity. The asymptotic shape of the level sets of the density is the common shape of the level sets of the intensity. For light-tailed densities, the asymptotic shape of the level sets of the density is the limit shape of the sample clouds. This paper investigates how the shape changes as the rate of decrease of the tails is varied while the copula of the distribution is preserved. Four cases are treated: a change from light tails to light tails, from heavy to heavy, heavy to light and light to heavy tails. Keywords Asymptotic dependence • Copula • Level set • Multivariate extremes • Risk • Sample cloud • Shape • Unimodal density
Mathematics Subject Classification (2010): Primary 60G70; Secondary 60E05, 60D05 G. Balkema () Department of Mathematics, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands e-mail: [email protected] P. Embrechts Department of Mathematics, ETH Zurich, Raemistrasse 101, 8092 Zurich, Switzerland e-mail: [email protected] N. Nolde Department of Statistics, University of British Columbia, 3182 Earth Sciences Building, 2207 Main Mall, Vancouver, BC, Canada V6T 1Z4 e-mail: [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 3, © Springer-Verlag Berlin Heidelberg 2013
43
44
G. Balkema et al.
1 Introduction Sample clouds evoke densities rather than distribution functions. Here a sample cloud is a finite set of independent observations from a multivariate distribution, treated as a geometric object, such as the set of points on a computer screen for a bivariate sample. Shape is important, the precise scale not. The classic models such as the multivariate Gaussian distribution and the Student t distributions have continuous unimodal densities, provided the distribution is nondegenerate. These densities are determined by a bounded set, the ellipsoid which describes the shape of the level sets of the density, and by the rate of decrease. In risk analysis one is interested in extreme observations, and it is the asymptotic shape of the level sets and the rate of decrease of the tails of the density that are important. Let us illustrate these two components with a few simple examples. A homothetic density has all level sets of the same shape, scaled copies of some given set. It is completely determined by the set and by the density generator which determines the decay along any ray. Altering the density on compact sets does not affect the asymptotic behaviour. So assume that the density is asymptotic to a homothetic density and impo
Data Loading...
