Traditionally, stochastic dominance is defined using the cumulative distribution function(CDF). But sometimes, the CDF is not easily to be obtained. For example, the generalized noncentral Chi-square distribution (known as the probability distribution of quadratic normal random variables) has a very complex CDF formula but a concise moment-generating function(MGF).
How to judge the stochastic dominance (mainly first and second order) of such random variables with MGFs?
For a random variable (r.v.) $X$, let $F_X$ and $M_X$ denote, respectively, the cdf and mgf of $X$, so that $$F_X(x)=P(X\le x)$$ for real $x$ and $$M_X(t)=Ee^{tX}$$ for real $t$.
Recall that a r.v. $Y$ dominates a r.v. $X$ (in the first order) if $F_X\ge F_Y$. Recall also that $Y$ dominates $X$ iff $Y_1\ge X_1$ for some copies in distribution $X_1$ and $Y_1$ of $X$ and $Y$, respectively.
It follows that, if $Y$ dominates $X$, then $M_Y\ge M_X$ on $[0,\infty)$.
However, the reverse implication is false. For instance, if $X=0$ and $P(Y=1)=P(Y=-1)=1/2$, then $M_X(t)=1\le\cosh t=M_Y(t)$ for all real $t$, but $Y$ does not dominate $X$ (and $X$ does not dominate $Y$).
So, in general the mgf domination does not imply anything about the cdf domination.
