Monotone likelihood ratio

Summary

Consider a family of densities indexed by a real parameter. It has a monotone likelihood ratio in a statistic $T$ when larger parameters make larger values of $T$ relatively more likely. The ratio of the densities at two parameter values is then a nondecreasing function of $T$. This property makes one-sided testing simple: tests that reject for large $T$ are optimal. One-parameter exponential families with an increasing natural parameter have this property.