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?
