Cutting levels of the winning probability relation of random variables pairwisely coupled by a same Frank copula

Abstract

We consider the winning probability relation associated with a set of (real-valued) random variables. Its computation requires knowledge of the marginal cumulative distribution functions of the random variables and of the bivariate cumulative distribution function of any couple of these random variables. Here, the copulas underlying these bivariate cumulative distribution functions are assumed to be identical and to belong to the parametric family of Frank copulas, although this most likely does not represent the real dependence and might even not be feasible at all; the Frank copula parameter should thus be regarded as a parameter of the method presented. This winning probability relation is then exploited to establish a strict partial order on the set of random variables. This is realized by computing an appropriate α-cut that yields a cycle-free crisp relation, resulting in a strict partial order upon computing its transitive closure. For any given Frank copula, we focus on finding the lowest possible value of α, called (minimal) cutting level, such that cutting the winning probability relation strictly above that value results in a crisp relation that is free from cycles of length m, with m∈{3,4,…}, irrespective of the marginal cumulative distribution functions of the random variables. We are able to give closed-form expressions for these cutting levels: (1) for all m when the copula is either the independence copula or one of the two Fréchet-Hoeffding bounds; (2) for any Frank copula when m=4 or when m tends to infinity. The relationship between the cutting level, the cycle length and the Frank copula parameter is discussed.