A modified version of stochastic dominance involving dependence

Abstract

Stochastic orders are mathematical methods allowing the comparison of random quantities. Probably the most usual one is stochastic dominance, which is based on the comparison of univariate cumulative distribution functions. Although it has been commonly applied, it does not consider the dependence between the random variables. This paper introduces a new stochastic order that slightly modifies stochastic dominance preserving its philosophy but taking into account the dependence between the random variables. This new stochastic order is based on the comparison of the cumulative distribution functions of the differences of the random variables, and it is closely related to regret theory. Since it uses the joint distribution, the copula gathering the dependence plays a crucial role. We present a theoretical study of this new stochastic order, delving into its connection with regret theory, investigating the role of the copula that links the random variables and establishing some connections with stochastic dominance.