Estimation of multivariate tail quantities

For d ≥ 2 risk variables, three methods have been proposed to estimate the multivariate tail quantities, including multivariate tail probabilities, tail dependence functions and tail quantile sets. The methods are based on weak assumptions on the joint tails of the copulas of the d variables. The first method is developed based on the tail expansion of copula along different directions to the joint upper or lower orthant. The latter two methods are based on the asymptotic expansion of a family of tail-weighted functions defined from the copula. Extensive simulation experiments are conducted to evaluate and compare the three methods under different scenarios. The simulation results show that the methods yield accurate estimates of the tail quantities and effectively distinguish the tail properties, such as reflection asymmetry, permutation asymmetry, and heterogeneous tail dependence. One data example is presented to illustrate the applicability of the proposed methods as inference and diagnostic tools.