On continuity of Chatterjee's rank correlation and related dependence measures

2025/03/14 by Ansari, Jonathan, Fuchs, Sebastian · 5 citations
#FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2503.11390

Abstract

While measures of concordance -- such as Spearman's rho, Kendall's tau, and Blomqvist's beta -- are continuous with respect to weak convergence, Chatterjee's rank correlation xi recently introduced in Azadkia and Chatterjee [5] does not share this property, causing drawbacks in statistical inference as pointed out in Bücher and Dette [7]. As we study in this paper, xi is instead weakly continuous with respect to conditionally independent copies -- the Markov products. To establish weak continuity of Markov products, we provide several sufficient conditions, including copula-based criteria and conditions relying on the concept of conditional weak convergence in Sweeting [36]. As a consequence, we also obtain continuity results for xi and related dependence measures and verify their continuity in the parameters of standard models such as multivariate elliptical and l1-norm symmetric distributions.

Cited by

• An MTP₂ property for conditional distributions
• On the exact region between Chatterjee's rank correlation and Spearman's footrule
• On exact regions between measures of concordance and Chatterjee's rank correlation for lower semilinear copulas
• A dimension reduction for extreme types of directed dependence
• Measures of association for approximating copulas

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