# Is there a robust test for stochastic dominance between two random variables?

I am trying to compare the errors from two statistical models in order to give evidence to one being "better" in terms of lower prediction error than the other.

To formalize this, I thought that a test of stochastic dominance between two collections of random variables (the OOS errors) would be a good idea. Ideally the null hypothesis would be :

$$\mathbb{H}_0: F(x) \ge G(x) \forall x \in \mathbb{R}$$

I have found resources pointing me to the Kruskal-Wallis test, but unfortunately cannot seem to find a paper explicitly stating and proving one of these (or similar) null hypotheses. Many sources I check simply state that the null is that the medians of the two distributions differ, but this is not what I want to check. Any help is appreciated.

• Since you have two samples, I believe the Mann-Whitney test would work for you: en.wikipedia.org/wiki/Mann%E2%80%93Whitney_U_test. Look at the first sentence under "Assumptions and formal statement of hypotheses". – jbowman Jan 12 '18 at 19:42
• It seems to be a paired sample, right? – Michael M Jan 12 '18 at 20:02