Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-23T21:54:06.359Z Has data issue: false hasContentIssue false
  • English
  • Français

Relations Between Hidden Regular Variation and the Tail Order of Copulas

Part of: Multivariate analysis

Published online by Cambridge University Press:  30 January 2018

Lei Hua ,
Harry Joe  and
Haijun Li
Lei Hua*
Affiliation:
Northern Illinois University
Harry Joe*
Affiliation:
University of British Columbia
Haijun Li*
Affiliation:
Washington State University
*
∗ Postal address: Division of Statistics, Northern Illinois University, DeKalb, IL, 60115, USA. Email address: hua@math.niu.edu
∗∗ Postal address: Department of Statistics, University of British Columbia, Vancouver, BC, V6T1Z4, Canada. Email address: harry.joe@ubc.ca
∗∗∗ Postal address: Department of Mathematics, Washington State University, Pullman, WA, 99164, USA. Email address: lih@math.wsu.edu
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.

We study the relations between the tail order of copulas and hidden regular variation (HRV) on subcones generated by order statistics. Multivariate regular variation (MRV) and HRV deal with extremal dependence of random vectors with Pareto-like univariate margins. Alternatively, if one uses a copula to model the dependence structure of a random vector then the upper exponent and tail order functions can be used to capture the extremal dependence structure. After defining upper exponent functions on a series of subcones, we establish the relation between the tail order of a copula and the tail indexes for MRV and HRV. We show that upper exponent functions of a copula and intensity measures of MRV/HRV can be represented by each other, and the upper exponent function on subcones can be expressed by a Pickands-type integral representation. Finally, a mixture model is given with the mixing random vector leading to the finite-directional measure in a product-measure representation of HRV intensity measures.

Keywords

Multivariate regular variationtail dependenceupper exponent functiontail order functionintermediate tail dependence

MSC classification

Primary: 62H20: Measures of association (correlation, canonical correlation, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 37 - 57
Copyright
© Applied Probability Trust 

Footnotes

Supported by a start-up grant at Northern Illinois Universit.y

Supported by an NSERC Canada Discovery grant.

Supported by NSF grants CMMI 0825960 and DMS 1007556.

References

Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press.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
Hashorva, E. (2010). On the residual dependence index of elliptical distributions. Statist. Prob. Lett. 80, 1070–1078.CrossRefGoogle Scholar
Hua, L. and Joe, H. (2011). Tail order and intermediate tail dependence of multivariate copulas. J. Multivariate Anal. 102, 1454–1471.Google Scholar
Hua, L. and Joe, H. (2012a). Tail comonotonicity and conservative risk measures. ASTIN Bull. 42, 601–629.Google Scholar
Hua, L. and Joe, H. (2012b). Tail comonotonicity: properties, constructions, and asymptotic additivity of risk measures. Insurance Math. Econom. 51, 492–503.CrossRefGoogle Scholar
Hua, L. and Joe, H. (2013). Intermediate tail dependence: a review and some new results. In Stochastic Orders in Reliability and Risk (Lecture Notes Statist. 208), Springer, New York, pp. 291–311.CrossRefGoogle Scholar
Jaworski, P. (2004). On uniform tail expansions of bivariate copulas. Appl. Math. (Warsaw) 31, 397–415.CrossRefGoogle Scholar
Jaworski, P. (2006). On uniform tail expansions of multivariate copulas and wide convergence of measures. Appl. Math. (Warsaw) 33, 159–184.Google Scholar
Joe, H. (1997). Multivariate Models and Dependence Concepts (Monogr. Statist. Appl. Prob. 73). Chapman & Hall, London.Google Scholar
Joe, H., Li, H. and Nikoloulopoulos, A. K. (2010). Tail dependence functions and vine copulas. J. Multivariate Anal. 101, 252–270.Google Scholar
Klüppelberg, C., Kuhn, G. and Peng, L. (2008). Semi-parametric models for the multivariate tail dependence function—the asymptotically dependent case. Scand. J. Statist. 35, 701–718.Google Scholar
Ledford, A. W. and Tawn, J. A. (1996). Statistics for near independence in multivariate extreme values. Biometrika 83, 169–187.Google Scholar
Li, H. and Sun, Y. (2009). Tail dependence for heavy-tailed scale mixtures of multivariate distributions. J. Appl. Prob. 46, 925–937.CrossRefGoogle Scholar
Maulik, K. and Resnick, S. (2004). Characterizations and examples of hidden regular variation. Extremes 7, 31–67.CrossRefGoogle Scholar
Mitra, A. and Resnick, S. (2010). Hidden regular variation: Detection and estimation. Preprint. Available at http:/arxiv.org/abs/1001.5058v2.Google Scholar
Mitra, A. and Resnick, S. I. (2011). Hidden regular variation and detection of hidden risks. Stoch. Models 27, 591–614.CrossRefGoogle Scholar
Nikoloulopoulos, A. K., Joe, H. and Li, H. (2009). Extreme value properties of multivariate t copulas. Extremes 12, 129–148.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes (Appl. Prob. 4). Springer, New York.CrossRefGoogle Scholar
Resnick, S. I. (2007). Heavy-tail Phenomena. Springer, New York.Google Scholar
Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231.Google Scholar