Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top 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?

share|cite|improve this question
up vote 1 down vote accepted

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.

share|cite|improve this answer

A simple example might be considering discrete pdfs like,

\begin{gather} f(x)=\begin{cases} 0.1\quad x=0\\ 0.1\quad x=5\\ 0.9\quad x=10 \end{cases} \end{gather} and

\begin{gather} g(x)=\begin{cases} 0.05 \quad x=0\\ 0.05 \quad x=5\\ 0.9\quad x=10 \end{cases} \end{gather}

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.