# Monotone Likelihood Ratio and Stochastic Dominance

neither monotone hazard rates nor stochastic dominance imply the MLRP

what is an example? Is there any necessary and sufficient condition for stochastic dominance?

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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.

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A simple example might be considering discrete pdfs like,