Robust stochastic orders and applications to elliptical distributions under parametric ambiguity

Stochastic orders provide a powerful method for comparing random variables based on their distributions, extensively studied by many scholars. However, in reality, it is often difficult to obtain the true distribution of a random variable due to data limitations. To address this challenge, we introduce a novel concept of robust stochastic order, where the distribution of the random variable is estimated within a family of distributions. Specifically, we study the relationships of several important robust stochastic orders and smooth generators of robust integral stochastic orders, which generalize the corresponding results of classical stochastic orders. The main focus of this study is on various robust stochastic orderings of elliptical distributions under parametric ambiguity, including robust usual stochastic order, robust convex order, robust supermodular order, robust directionally convex order, robust componentwise convex order, and so on. As an application grounded in expected utility theory, we present some examples, providing more reasonable decision-making guidances for risk-averse investors.