= Monotone test
An upper-tail monotone test in a real statistic $T$ has rejection probability $\phi=\mathbf1_{T>c}+\eta\mathbf1_{T=c}$, with $0\le\eta\le1$; 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 $\phi-\phi'$ and $r-k$ agree away from $T=c$, and $r=k$ at a boundary with positive mass. Therefore $E_{\theta_1}(\phi-\phi')\ge kE_{\theta_0}(\phi-\phi')$. 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.
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