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On the estimation of Spearman’s rho and related tests of independence for possibly discontinuous multivariate data. (English) Zbl 1359.62210

Summary: 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 J.-F. Quessy [Commun. Stat., Theory Methods 38, No. 19, 3510–3531 (2009; Zbl 1177.62081)] and M. Mesfioui and J.-F. Quessy [J. Multivariate Anal. 101, No. 10, 2398–2410 (2010; Zbl 1207.62129)]. 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 C. Genest and B. Rémillard [Test 13, No. 2, 335–370 (2004; Zbl 1069.62039)]. 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.

MSC:

62H20 Measures of association (correlation, canonical correlation, etc.)
62H15 Hypothesis testing in multivariate analysis
62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
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