Monotone Likelihood Ratio and Stochastic Dominance < says neither monotone hazard rates nor stochastic dominance imply the MLRP what is an example? Is there any necessary and sufficient condition for stochastic dominance?

There are no doubt many examples, but for example consider two distributions on the interval $[0,1]$ one bimodal at $\frac{1}{4}$ and $\frac{3}{4}$ and the other unimodal at $\frac{1}{2}$: $Y$ with density $f(x)=1 - \cos(4\pi x)$ and $Z$ with density $g(x)=1 - \cos(2 \pi x)$.

Then $\Pr(Y \le x ) \gt \Pr(Z \le x )$ in the open interval $(0,1)$ [and equal elsewhere] so there is stochastic dominance.

But in this example the likelihood ratio $\frac{f(x)}{g(x)}$ is not monotone: it falls and then rises.