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 dependencemeasure 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 .
A new class of measures of bivariate tail dependence called tail concordance measures (TCMs) is proposed, which is defined as the limit of a measure of concordance of the underlying copula restricted to the tail region of interest. TCMs captures the extremal relationship between random variables not only along the diagonal but also along all angles weighted by a tail generating measure. Axioms of tail dependence measures are introduced, and TCMs are shown to characterize linear tail dependence measures . The infimum and supremum of TCMs over all generating measures are considered to investigate the issue of under- and overestimation of the degree of extreme co-movements. The infimum is shown to be attained by the classical tail dependence coefficient, and thus the classical notion always underestimates the degree of tail dependence. A formula for the supremum TCM is derived and shown to overestimate the degree of extreme co-movements. Estimators of the proposed measures are studied , and their performance is demonstrated in numerical experiments. For a fair assessment of tail dependence and stability of the estimation under small sample sizes, TCMs weighted over all angles are suggested, with tail Spearman's rho and tail Gini's gamma being interesting novel special cases of TCMs .
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