Karlin-Rubin theorem

Extension of the Neyman-Pearson lemma to classes with monotone likelihood ratio. Here’s the statement.

Suppose $\{P_{\theta }:\theta \in \Theta \subset \mathbb{R}\}$ has MLR in the sufficient statistic $T$ and consider testing $H_0:\theta \le {\theta }_0$ vs $H_1:\theta >{\theta }_0$. Then the uniformly most powerful test has the form

$$\varphi (X)=\{\begin{array}{ll}1,& T(X)>c,\\ \gamma ,& T(X)=c,\\ 0,& T(X)<c,\end{array}$$

where $\gamma$ and $c$ are chosen such that ${\sup }_{\theta \le {\theta }_0}{P}_{\theta }(\varphi (X)=1)=\alpha$. The main takeaway is that the test is a threshold test based on the statistic $T$.