## 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

### Citations:

Zbl 0459.62010; Zbl 0909.62052; Zbl 1360.62294
Full Text:

### References:

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