Kendall's tau and Spearman's rho for n-dimensional Archimedean copulas and their asymptotic properties

摘要

We derive formulas for the dependence measures tau(n) and rho((1))(n), rho((2))(n) for Archimedean n-copulas. These measures are n-dimensional analogues of the popular nonparametric dependence measures: Kendall's tau and Spearman's rho. For tau(n) we obtain two formulas, both involving integrals of univariate functions. The formulas for rho((1))(n), rho((2))(n) involve integrals of n-variate functions. We also obtain formulas for the three measures for copulas whose additive generators have completely monotone inverses. These formulas feature integrals of 2-variate functions (we use the Laplace transform). We study the asymptotic properties of the sequence (tau(n) (C-n)) and (rho((1))(n) (C-n)), (rho((2))(n) (C-n))for a sequence (C-n) of Archimedean copulas with a common additive generator. We also investigate the limit of this sequence, which is an infinite-dimensional copula on the Hilbert cube.