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 prior density $\pi$ and cutoff $a$, the posterior odds of $\theta >a$ against $\theta \le a$ are nondecreasing in $T$. To prove it, write the unnormalized masses as $A(t)$ and $B(t)$. For $t_2>t_1$, the difference $A(t_2)B(t_1)-A(t_1)B(t_2)$ is the double integral over $\theta >a\ge \eta$ of $\pi (\theta )\pi (\eta )[f_{\theta }(t_2)f_{\eta }(t_1)-f_{\theta }(t_1)f_{\eta }(t_2)]$. Each integrand is nonnegative by the likelihood-ratio ordering. Common factors in the full-data likelihood cancel from the odds, so the same argument works when $T$ is a statistic rather than the complete observation. Interpret zero denominators by extended odds whenever the posterior is defined.

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.

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