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

David Lee, Harry Joe and Pavel Krupskii

Journal of Nonparametric Statistics, 2018, vol. 30, issue 2, 262-290

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 $ {\hat \vartheta }_\alpha $ ϑˆα, for $ \alpha >0 $ α>0, 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 $ \zeta _\alpha $ ζα of $ \hat {\zeta }_\alpha =2-{\hat \vartheta }_\alpha $ ζˆα=2−ϑˆα as the sample size goes to infinity and show that (i) $ \zeta _\alpha $ ζα for $ \alpha =1 $ α=1 is a measure of central dependence with properties similar to Kendall's tau and Spearman's rank correlation, (ii) $ \zeta _\alpha $ ζα is a tail-weighted dependence measure for large α, and (iii) the limit as $ \alpha \to \infty $ α→∞ is the upper tail dependence coefficient. We obtain asymptotic properties for the rank-based measure $ {\hat \zeta }_\alpha $ ζˆα and estimate tail dependence coefficients through extrapolation on $ {\hat \zeta }_\alpha $ ζˆα. A data example illustrates the use of the new dependence measures for tail inference.

Date: 2018
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DOI: 10.1080/10485252.2017.1407414

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