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

摘要

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 upsilon 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 zeta alpha = 2 - upsilon 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 -infinity is the upper tail dependence coefficient. We obtain asymptotic properties for the rank-based measure zeta alpha and estimate tail dependence coefficients through extrapolation on zeta alpha. A data example illustrates the use of the new dependence measures for tail inference.