# Monotone Likelihood Ratio for simple null vs composite alternative

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 $$H_1:\theta\lt \theta_0$$.

• 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 $H_1':\theta<\theta_0$. Mar 26 at 17:03