I'm reading the paper Invariant directional ordering, and I'm confused by two propositions.
The basic definition is
If it holds, then
My first question is, in proposition $(1.6)$, shouldn't it be $E_F\psi(X) \ge E_G\psi(X)$, given $F(x) \ge G(x)$?
And for the second one:
. Shouldn't the distribution function of $\psi(X)$ be $F(\psi^{-1})$? Why it says the distribution is $F\psi^{-1}$?
No, $(1.6)$ is correct as it stands. $F(x)\ge G(x)$ means that $X$ (the variable corresponding to $F$) tends to be smaller than $Y$ (the variable corresponding to $G$), since $F$ and $G$ specify the probability for the variable to be less than $x$, and this probability is always larger for $X$ than for $Y$.
For the second question, it’s not clear to me what you mean by $F(\psi^{-1})$; this is not a standard notation. The standard meaning of $F\psi^{-1}$ is composition of functions, so $F\psi^{-1}$ is the function $F\psi^{-1}:\mathbb R\to\mathbb R,x\mapsto F(\psi^{-1}(x))$, and this is indeed the cumulative distribution function of $\psi(X)$. Perhaps this is also what you mean by $F(\psi^{-1})$, but that is not how this composition is conventionally denoted. Another conventional notation for it is $F\circ\psi^{-1}$.