monotone likelihood ratio

Definition

A family of densities parameterized by real values $\theta$, $\mathcal{P}=\{P_{\theta }:\theta \in \Theta \subset \mathbb{R}$} has monotone likelihood ratio in $T(X)$ (some statistic of the data) if for any ${\theta }_1<{\theta }_2$, the likelihood ratio $\text{d}{P}_{{\theta }_2}(x)/\text{d}{P}_{{\theta }_1}(x)$ is well-defined and is a monotonically increasing function of $T(x)$.

This might some an obscure property, but many families of distributions have the monotone likelihood ratio property. For eg:

• One parameter exponential families
• Exponential and double exponential distributions
• Logistic distributions
• Uniform distributions
• Hypergeometric distributions

The MLR property is important in hypothesis testing: The Karlin-Rubin theorem extends the Neyman-Pearson lemma to MLR families. This makes the NP lemma significantly more practical, as testing point nulls vs point composites is rarely done in practice.