On the estimation of Spearman’s rho and related tests of independence for possibly discontinuous multivariate data

Tie-corrected versions of Spearman’s rho are often used to measure the dependence in a pair of non-continuous random variables. Multivariate extensions of this coefficient, and estimators thereof, have recently been proposed by Quessy (2009a) [23] and Mesfioui and Quessy (2010) [19]. Asymptotically equivalent but numerically much simpler estimators of the same coefficients are given here. Expressions are also provided for their limiting variance, thereby correcting errors in these authors’ papers. It is further shown that the Möbius decomposition of the multilinear extension (or checkerboard) copula leads to tie-corrected versions of dependence coefficients originally introduced by Genest and Rémillard (2004) [10]. These coefficients can be used to visualize dependence structures and to construct tests of mutual independence that can be more powerful than those based on tie-corrected versions of Spearman’s rho.