I was studying the monotone likelihood ratio property. I have a small query. We know that once a distribution has a MLR in $T(X)$ for $\theta$ then I can test one sided hypothesis of the form $H_0:\theta\ge \theta_0$ vs $H_1:\theta\lt \theta_0$. I just want to know whether I can use the test derived from this to test $H_0:\theta = \theta_0$ vs $H_1:\theta\lt \theta_0$.

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    $\begingroup$ This is a part of the Karlin-Rubin theorem. If there is MLR in $T$, then for testing $H_0:\theta=\theta_0$ against $H_1:\theta=\theta_1(<\theta_0)$, for every $\alpha\in (0,1)$, there exists a test of the form $\phi=\mathbf1_{T<c}+k\mathbf1_{T=c}$ with $E_{\theta_0}\phi=\alpha$. And that this same test $\phi$ is UMP size $\alpha$ for testing both $H_0$ and $H_0':\theta\ge \theta_0$ against $H_1':\theta<\theta_0$. $\endgroup$ Mar 26 at 17:03

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