Mean-risk tests of stochastic dominance

We propose a new approach to testing whether one random variable is stochastically non-dominated by another one. The tests compare mean-risk differences of two unknown distributions using independent samples. The test can be used for comparison of the coherent risk measures of the distributions, as well as to reject stochastic dominance relation of first, second, or higher order between the two distributions. We consider several law-invariant coherent measures of risk which are consistent with the stochastic dominance relation of first and higher order. Numerical comparisons with the Mann-Whitney test and with the F-test for comparison of variance are provided. The numerical study indicates that most of the mean-risk tests are more powerful than the Mann-Whitney test.