Monotone likelihood ratio

A one-parameter family has a monotone likelihood ratio in a statistic $T$ when $f(x\mid {\theta }_1)/f(x\mid {\theta }_0)$ is nondecreasing in $T(x)$ whenever ${\theta }_1>{\theta }_0$.

For a family with a monotone likelihood ratio in $T$, a level-$\alpha$ upper-tail test in $T$ is a uniformly most powerful test for the corresponding one-sided hypothesis.

For one observation from the logistic distribution, the size-$\alpha$ test of $H_0:\theta \le 0$ against $H_1:\theta >0$ rejects when