Semi-nonparametric test of second degree stochastic dominance with respect to a function

In an expected utility framework, assuming a decision maker operates under utility k ( ⋅ | θ ) , for two risky alternatives X and Y with respective distribution functions F and G , alternative X is said to dominate alternative Y with respect to k ( ⋅ | θ ) if ∫ − ∞ y [ F ( t ) − G ( t ) ] d k ( t | θ ) ≤ 0 for all y . Utilizing the empirical distribution functions of F and G , a statistical test is presented to test the null hypothesis of indifference between X and Y given k ( ⋅ | θ ) against the hypothesis that X dominates Y with respect to k ( ⋅ | θ ) . This is a large sample testing application of stochastic dominance with respect to a function. The asymptotic distribution of the test statistic associated with the null hypothesis given a sub-set of the utility function parameter space is developed. Based on large sample rejection regions, the hypothesis of preference of one alternative over another is demonstrated with an empirical example.