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On the properties of some nonparametric concordance measures in the discrete case. (English) Zbl 1135.60303

Summary: It is shown here that Kendall’s \(\tau\) and Spearman’s \(\rho\) are monotone with respect to the concordance ordering of pairs of discrete as well as continuous random variables. This extends and completes results of A. H. Tchen [Ann. Probab. 8, 814–827 (1980; Zbl 0459.62010)] It is also shown that various relationships between Kendall’s \(\tau\) and Spearman’s \(\rho\) mentioned in [R. B. Nelsen, An introduction to copulas. Lecture Notes in Statistics 139. New York: Springer (1999; Zbl 0909.62052)] remain valid for discrete variables. In particular, a result of P. Capéraà and C. Genest [ J. Nonparametric Stat. 2, No. 2, 183–194 (1993; Zbl 1360.62294)] is extended to the case of discrete random pairs. Finally, an analytic expression is given for the most extreme values of Kendall’s \(\tau\) and Spearman’s \(\rho\) associated with discrete uniform variates.

MSC:

60E15 Inequalities; stochastic orderings
60E05 Probability distributions: general theory
62G30 Order statistics; empirical distribution functions
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References:

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