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 ${hatvartheta}_alpha$, for $alpha>0$, of the extremal coefficient, based on a transform of the absolute difference of the $alpha$ power of the ranks. In the case of general bivariate copulas, we obtain the probability limit $zeta_alpha$ of $hat{zeta}_alpha=2-{hatvartheta}_alpha$ as the sample size goes to infinity, and show that (i) $zeta_alpha$ for $alpha=1$ is a measure of central dependence with properties similar to Kendall's tau and Spearman's rank correlation, (ii) $zeta_alpha$ is a tail-weighted dependence measure for large $alpha$, and (iii) the limit as $alpha oinfty$ is the upper tail dependence coefficient. We obtain asymptotic properties for the rank-based measure ${hatzeta}_alpha$, and estimate tail dependence coefficients through extrapolation on ${hatzeta}_alpha$. A data example illustrates the use of the new dependence measures for tail inference.