Monotone test

An upper-tail monotone test in a real statistic $T$ has rejection probability $\varphi =1_{T>c}+\eta 1_{T=c}$, with $0\le \eta \le 1$; the constant tests correspond to infinite cutoffs. For a family with positive densities on a common support and a monotone likelihood ratio in $T$, its power function can cross that of an arbitrary competing test only from below to above as the parameter increases. If $r=f_{{\theta }_1}/f_{{\theta }_0}$ and ${\theta }_1>{\theta }_0$, choose a positive number $k$ between the ratios on opposite sides of the cutoff. The signs of $\varphi -{\varphi }^{\prime }$ and $r-k$ agree away from $T=c$, and $r=k$ at a boundary with positive mass. Therefore $E_{{\theta }_1}(\varphi -{\varphi }^{\prime })\ge k{E}_{{\theta }_0}(\varphi -{\varphi }^{\prime })$. In particular strict positivity at ${\theta }_0$ propagates to ${\theta }_1$. Positivity of the densities matters for this strict conclusion; with changing supports only a weak conclusion may remain.