Suppose, there are two distributions over [0, 1] such that $F(x)=1-G(1-x)$ for all $x\in[0,1]$ and monotone likelihood ratio holds, i.e. $\frac{f(x)}{g(x)}$ is increasing in $x$. Does that necessarily imply that both $f(x)$ and $g(x)$ are monotonic? It seems to be the case, but I cannot see formally why. Thanks.
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Thus $g(x)=f(1-x)$ and $g(1-x)/g(x)$ is increasing. Why should this imply that $g$ is monotonic? – Did Nov 09 '15 at 15:13
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It does not. But MLR property implies that F FOSDs G. I cannot construct cdfs such that $F(x)<=G(x)$, $F(x)=1-G(1-x)$, and at the same time both functions have non-monotone derivatives. Or I am just missing something – fencer Nov 09 '15 at 15:32
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Actually, I can't construct such cdfs without violating MLR property. – fencer Nov 09 '15 at 16:21