15 - Transitive comparison of random variables

Abstract

This chapter presents a general framework for describing the transitivity of probabilistic relations. It discusses a procedure for expressing the pairwise comparison of independent random variables in terms of a probabilistic relation. The transitivity of this relation is studied both in general and within popular parameterized families of distributions. The chapter also explains how a copula can be entered into this comparison procedure and how this affects the transitivity of the probabilistic relation. A collection of generalized dice together with the probabilistic relation made of the winning probabilities among these dice is called a discrete dice model. The chapter generalizes the discrete dice model to collections of independent discrete or continuous random variables with arbitrary probability distributions and illustrates that the generated probabilistic relations that provide an alternative to the concept of stochastic dominance are still dice-transitive. A further generalization consists of allowing the random variables to be dependent. For the pairwise comparison of such random variables, the two-dimensional marginal distributions are needed. A copula is called stable if it coincides with its survival copula.