# Lecture 13. C) Karlin-Rubin Theorem

The Karlin-Rubin theorem states that the UMP level $\alpha$ test exists for one-sided testing,
$H_{0}:\theta\leq\theta_{0}$ vs. $H_{1}:\theta\gt \theta_{0}$
if $f\left(\left.X\right|\theta\right)$ belongs to the exponential family and $\omega\left(\theta\right)$ is monotone (i.e., the pdf/pmf satisfies the monotone likelihood ratio property).
In this case, we reject $H_{0}$ when $\sum_{i=1}^{n}t\left(X_{i}\right)$ is large.