Karlin Rubin Test for Monotonically Non Increasing Likelihood Ratio What if the likelihood ratio of karlin rubin test is monotonically nonincreasing (instead of nondecreasing), can karlin rubin test still provide UMP test for one sided alternative?. If yes, then what would the UMP test looks like?.
 A: The likelihood ratio only has to be monotone for Karlin-Rubin theorem to apply. So there is no problem if the ratio is monotone non-increasing. Only the critical region is reversed.
Let $\mathcal F=\{f_{\theta}:\theta\in \Omega\}$ be a family of identifiable distributions.
Then $\mathcal F$ has monotone likelihood ratio (MLR) if $r(x)=\frac{f_{\theta_2}(x)}{f_{\theta_1}(x)}$ is monotone in some statistic $T(x)$ for $\theta_2>\theta_1$ and for every $x$ for which $f_{\theta_1}(x)>0$ or $f_{\theta_2}(x)>0$.
If $r(x)$ is non-decreasing in $T(x)$, then $\mathcal F$ has MLR in $T(x)$.
If $r(x)$ is non-increasing in $T(x)$, then $\mathcal F$ has MLR in $-T(x)$.
So suppose $\mathcal F$ has MLR in $-T(x)$. By Karlin-Rubin theorem, for testing $H_0:\theta=\theta_0$ against $H_1:\theta=\theta_1 (>\theta_0)$, a UMP size $\alpha$ test for every $\alpha \in (0,1)$ is of the form
$$\varphi(x)=\begin{cases}1 &,\text{ if } T(x)<c 
\\ \gamma &,\text{ if } T(x)=c 
\\ 0 &,\text{ if } T(x)>c \end{cases}$$
where $c$ and $\gamma \in[0,1]$ are such that $E_{\theta_0} \varphi(X)=\alpha$.
