Tail-weighted dependence measures with limit being the tail dependence coefficient

ABSTRACT

For bivariate continuous data, measures of monotonic dependence are based on the rank transformations of the two variables. For bivariate extreme value copulas, there is a family of estimators , for , of the extremal coefficient, based on a transform of the absolute difference of the α power of the ranks. In the case of general bivariate copulas, we obtain the probability limit of as the sample size goes to infinity and show that (i) for is a measure of central dependence with properties similar to Kendall's tau and Spearman's rank correlation, (ii) is a tail-weighted dependence measure for large α, and (iii) the limit as is the upper tail dependence coefficient. We obtain asymptotic properties for the rank-based measure and estimate tail dependence coefficients through extrapolation on . A data example illustrates the use of the new dependence measures for tail inference.

KEYWORDS:

• Copula
• extremal coefficient
• monotone dependence
• tail order
• tail-weighted dependence

Acknowledgements

The authors would like to thank the anonymous referees for their useful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

David Lee http://orcid.org/0000-0001-9957-233X

Notes

1 We remark that many of these methods were initially designed to estimate the Pickands dependence function , . Because of the homogeneity property of A, we have .

2 We observe that the rate of growth of the asymptotic variance is typically between and in the range of for copulas without tail dependence.

Additional information

Funding

This research has been supported by UBC's Four Year Doctoral Fellowship, NSERC Discovery Grant 8698, a Collaborative Research Team grant for the project: Copula Dependence Modelling: Theory and Applications of the Canadian Statistical Sciences Institute, and the King Abdullah University of Science and Technology.