Conditional Precedence Orders for Stochastic Comparison of Random Variables

TL;DR

This paper examines the limitations of existing stochastic orders for comparing random variables, especially regarding transitivity and dependence, and proposes variations to improve their reliability in decision-making contexts.

Contribution

The paper introduces modified stochastic precedence orders to overcome transitivity issues and dependence considerations in stochastic comparisons.

Findings

Standard stochastic orders lack connex property and dependence consideration.

Stochastic precedence order can lead to misleading conclusions.

Proposes variations of the order to address these issues.

Abstract

Most of the stochastic orders for comparing random variables, considered in the literature, are afflicted with two main drawbacks: (i) lack of connex property and (ii) lack of consideration of any dependence structure between the random variables. Both these drawbacks can be overcome at the cost of transitivity with the stochastic precedence order, which may seem to be a good choice in particular when only two random variables are under consideration, a situation where the question of transitivity does not arise. In this paper, we show that even under such favorable conditions, stochastic precedence order may direct to misleading conclusion in certain situations and develop variations of the order to address the phenomenon.