Multivariate directional tail-weighted dependence measures

We propose a new family of directional dependence measures for multivariate distributions. The family of dependence measures is indexed by $\alpha \ge 1$. When $\alpha =1$, they measure the strength of dependence along different paths to the joint upper or lower orthant. For $\alpha$ large, they become tail-weighted dependence measures that put more weight in the joint upper or lower tails of the distribution. As $\alpha \to \infty$, we show the convergence of the directional dependence measures to the multivariate tail dependence function and characterize the convergence pattern with an asymptotic expansion. This expansion leads to a method to estimate the multivariate tail dependence function using weighted least square regression. We develop rank-based sample estimators for the tail-weighted dependence measures and establish their asymptotic distributions. The practical utility of the tail-weighted dependence measures in multivariate tail inference is further demonstrated through their application to a financial dataset.