Let ${X_1}^-,...,{X_n}^-,{X_1}^+,...,{X_n}^+$ be real random variables with the following stochastic dominance property:
$P({X_1}^->a_1,...,{X_n}^->a_n)\leq P({X_1}^+>a_1,...,{X_n}^+>a_n)$ for all $a \in \mathbb R^n$.
Now let $h:\mathbb R^n \rightarrow \mathbb R^+$ be strictly increasing in every component.
Does then
$E[h({X_1}^-,...,{X_n}^-)]\leq E[h({X_1}^+,...,{X_n}^+)]$ hold?
Ideas: Using the definition of expectation brought me to a sub-problem that I asked here: In an inequality of integrals, can I multiply both integrands with the same non-negative, monotonous function? However, maybe that was already a step in the wrong direction, thus here the original problem.
Thanks in advance!
