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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
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Abstract

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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.

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