A new class of tail dependence measures and their maximization

A new class of measures of bivariate tail dependence is proposed, which is defined as a limit of a measure of concordance of the underlying copula restricted to the tail region of interest. The proposed tail dependence measures include tail dependence coefficients as special cases, but capture the extremal relationship between random variables not only along the diagonal but also along all the angles weighted by the so-called tail generating measure. As a result, the proposed tail dependence measures overcome the issue that the tail dependence coefficients underestimate the extent of extreme co-movements. We also consider the so-called maximal and minimal tail dependence measures, defined as the maximum and minimum of the tail dependence measures among all tail generating measures for a given copula. It turns out that the minimal tail dependence measure coincides with the tail dependence coefficient, and the maximal tail dependence measure overestimates the degree of extreme co-movements. We investigate properties, representations and examples of the proposed tail dependence measures, and their performance is demonstrated in a series of numerical experiments. For fair assessment of tail dependence and stability of estimation under small sample size, we support the use of tail dependence measures weighted over all angles compared with maximal and minimal ones.