Remarks on equality of two distributions under some partial orders

TL;DR

This paper investigates conditions under which two random variables or vectors can be considered stochastically equal when ordered by convex or supermodular relations, including multivariate cases.

Contribution

It provides new conditions for stochastic equality under convex and supermodular orderings, extending results to multivariate distributions.

Findings

Conditions for stochastic equality under convex ordering

Conditions for stochastic equality under supermodular ordering

Multivariate extensions of these conditions

Abstract

In this note we establish some appropriate conditions for stochastic equality of two random variables/vectors which are ordered with respect to convex ordering or with respect to supermodular ordering. Multivariate extensions of this result are also considered.