Bart De Schuymer

A fuzzy approach to stochastic dominance of random variables

The discrete dice model essentially amounts to comparing discrete uniform probability distributions and generates reciprocal relations that exhibit a particular type of transitivity called dice-transitivity. In this contribution, this comparison method is extended and applied to general probability distributions and the generated reciprocal relations are shown to generalize the concept of stochastic dominance. For a variety of parametrized probability distributions, we analyse the transitivity properties of these reciprocal relations within the framework of cycle-transitivity. The relationship between normal probability distributions and the different types of stochastic transitivity is emphasized.

Transitive comparison of random variables

A general framework for describing the transitivity of probabilistic relations is presented. A procedure is established 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. It is shown how a copula can be entered into this comparison procedure and how this affects the transitivity of the probabilistic relation.