Monotone Likelihood Ration (mlr)

Definition

Let $L(\theta ,\mathbf{x})$ : Likelihood function

If $\frac{L({\theta }_1,\mathbf{x})}{L({\theta }_2,\mathbf{x})},\forall {\theta }_1<{\theta }_2$ is a monotone function of $y=u(\mathbf{x})$

Then we say $L(\theta ,\mathbf{x})$ has monotone likelihood ratio (mlr) in the statistic $y=u(\mathbf{x})$