On Concordance Measures for Discrete Data and Dependence Properties of Poisson Model

Abstract

We study Kendall's tau and Spearman's rho concordance measures for discrete variables. We mainly provide their best bounds using positive dependence properties. These bounds are difficult to write down explicitly in general. Here, we give the explicit formula of the best bounds in a particular Fréchet space in order to understand the behavior of the ranges of these measures. Also, based on the empirical copula which is viewed as a discrete distribution, we propose a new estimator of the copula function. Finally, we give useful dependence properties of the bivariate Poisson distribution and show the relationship between parameters of the Poisson distribution and both tau and rho.