# Which form of stochastic dominance is tested for using the Mann-Whitney U hypothesis test?

The Mann–Whitney U test has the following interpretation reported on wikipedia:

[...] is a nonparametric test of the null hypothesis that, for randomly selected values X and Y from two populations, the probability of X being greater than Y is equal to the probability of Y being greater than X.

I had intuitively taken this to mean "one variable tends to be larger than the other", but I recently encountered different notions of stochastic dominance including statewise dominance, first order dominance, second-order dominance, third-order dominance, and a less-specific notion of higher-order stochastic dominance.

Statewise stochastic dominance:

Random variable A is statewise dominant over random variable B if A gives at least as good a result in every state (every possible set of outcomes), and a strictly better result in at least one state.

First-order stochastic dominance:

Random variable A has first-order stochastic dominance over random variable B if for any outcome x, A gives at least as high a probability of receiving at least x as does B, and for some x, A gives a higher probability of receiving at least x. In notation form, $$P[A\geq x]\geq P[B\geq x]P[A\geq x]\geq P[B\geq x]$$ for all $$x$$, and for some $$x$$, $$P[A>x]>P[B>x] P[A>x]>P[B>x]$$

Second-order stochastic dominance:

In terms of cumulative distribution functions $$F_{A}$$ and $$F_{B}$$, $$A$$ is second-order stochastically dominant over $$B$$ if and only if the area under $$F_{A}$$ from minus infinity to $$x$$ is less than or equal to that under $$F_{B}$$ from minus infinity to $$x$$ for all real numbers $$x$$, with strict inequality at some $$x$$; that is, $$\int _{-\infty }^{x}[F_{B}(t)-F_{A}(t)]\,dt\geq 0$$ for all $$x$$, with strict inequality at some $$x$$. Equivalently, $$A$$ dominates $$B$$ in the second order if and only if $${E} [u(A)]\geq \operatorname {E} [u(B)]$$ for all nondecreasing and concave utility functions $$u(x)$$.

Third-order stochastic dominance:

Let $$F_{A}$$ and $$F_{B}$$ be the cumulative distribution functions of two distinct investments $$A$$ and $$B$$. $$A$$ dominates $$B$$ in the third order if and only if $$\int _{-\infty }^{x}\int _{-\infty }^{z}[F_{B}(t)-F_{A}(t)]\,dt\,dz\geq 0{\text{ for all }}x$$ [AND] $${E} _{A}(x)\geq \operatorname {E} _{B}(x)$$, and there is at least one strict inequality. Equivalently, $$A$$ dominates $$B$$ in the third order if and only if $$\operatorname {E} _{A}U(x)\geq \operatorname {E} _{B}U(x)$$ for all nondecreasing, concave utility functions $$U$$ that are positively skewed (that is, have a positive third derivative throughout).

Higher-order stochastic dominance:

[Yet to be defined]

Which form of stochastic dominance is tested for using the Mann–Whitney U hypothesis test?

$$H_1: P(X_i > Y_j) \neq 0.5.$$