Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-04T19:18:58.319Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic independence for unimodal densities

Part of: Stochastic processes Distribution theory

Published online by Cambridge University Press:  01 July 2016

Guus Balkema  and
Natalia Nolde
Guus Balkema*
Affiliation:
University of Amsterdam
Natalia Nolde*
Affiliation:
ETH Zürich
*
∗ Postal address: Department of Mathematics, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands. Email address: [email protected]
∗∗ Postal address: Department of Mathematics and RiskLab, ETH Zürich, Raemistrasse 101, 8092 Zürich, Switzerland. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Asymptotic independence of the components of random vectors is a concept used in many applications. The standard criteria for checking asymptotic independence are given in terms of distribution functions (DFs). DFs are rarely available in an explicit form, especially in the multivariate case. Often we are given the form of the density or, via the shape of the data clouds, we can obtain a good geometric image of the asymptotic shape of the level sets of the density. In this paper we establish a simple sufficient condition for asymptotic independence for light-tailed densities in terms of this asymptotic shape. This condition extends Sibuya's classic result on asymptotic independence for Gaussian densities.

Keywords

Asymptotic independenceblunthomothetic densitylevel setskew normalstar shaped

MSC classification

Primary: 60G55: Point processes 60G70: Extreme value theory; extremal processes 62E20: Asymptotic distribution theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 2 , June 2010 , pp. 411 - 432
Copyright
Copyright © Applied Probability Trust 2010 

References

Arnold, B. C., Castillo, E. and Sarabia, J. M. (2008). Multivariate distributions defined in terms of contours. J. Statist. Planning Infer. 138, 4158–4171.CrossRefGoogle Scholar
Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika 83, 715–726.CrossRefGoogle Scholar
Balkema, A. A. and Embrechts, P. (2007). High Risk Scenarios and Extremes. European Mathematical Society, Zürich.CrossRefGoogle Scholar
Balkema, A. A., Embrechts, P. and Nolde, N. (2009). Sensitivity of the asymptotic behaviour of meta distributions. Preprint. Available at http://arxiv.org/abs/0912.5337v/.Google Scholar
Balkema, A. A., Embrechts, P. and Nolde, N. (2010). Meta densities and the shape of their sample clouds. J. Multivariate Anal. 101, 1738–1754.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
De Haan, L. and Ferreira, A. (2006). Extreme Value Theory. An Introduction. Springer, New York.CrossRefGoogle Scholar
Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA.Google Scholar
Fang, K. T., Kotz, S. and Ng, K. W. (1990). Symmetric Multivariate and Related Distributions. Chapman and Hall, London.CrossRefGoogle Scholar
Fernández, C., Osiewalski, J. and Steel, M. F. J. (1995). Modeling and inference with v-spherical distributions. J. Amer. Statist. Assoc. 90, 1331–1340.Google Scholar
Gnedin, A. V. (1993). On multivariate extremal processes. J. Multivariate Anal. 46, 207–213.CrossRefGoogle Scholar
Gnedin, A. V. (1994). On a best-choice problem with dependent criteria. J. Appl. Prob. 31, 221–234.CrossRefGoogle Scholar
Hashorva, E. (2005). Extremes of asymptotically spherical and elliptical random vectors. Insurance Math. Econom. 36, 285–302.CrossRefGoogle Scholar
Hult, H. and Lindskog, F. (2002). Multivariate extremes, aggregation and dependence in elliptical distributions. Adv. Appl. Prob. 34, 587–608.CrossRefGoogle Scholar
Kinoshita, K. and Resnick, S. I. (1991). Convergence of scaled random samples in R d . Ann. Prob. 19, 1640–1663.CrossRefGoogle Scholar
Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187.CrossRefGoogle Scholar
Lysenko, N., Roy, P. and Waeber, R. (2009). Multivariate extremes of generalized skew-normal distributions. Statist. Prob. Lett. 79, 525–533.CrossRefGoogle Scholar
Maulik, K., Resnick, S. and Rootzén, H. (2002). Asymptotic independence and a network traffic model. J. Appl. Prob. 39, 671–699.CrossRefGoogle Scholar
McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press.Google Scholar
Mikosch, T. (2006). Copulas: tales and facts. Extremes 9, 3–20.CrossRefGoogle Scholar
Osiewalski, J. and Steel, M. F. J. (1993). Robust Bayesian inference in ℓ q -spherical models. Biometrika 80, 456–460.Google Scholar
Ramos, A. and Ledford, A. (2009). A new class of models for bivariate Joint tails. J. R. Statist. Soc. B 71, 219–241.CrossRefGoogle Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
Resnick, S. I. (2002). Hidden regular variation, second order regular valuation and asymptotic independence. Extremes 5, 303–336.CrossRefGoogle Scholar
Sibuya, M. (1960). Bivariate extreme statistics. Ann. Inst. Statist. Math. 11, 195–210.CrossRefGoogle Scholar