def. let . To compare hypotheses:

  1. We define the Likelihood (Statistics) Ratio as:

Then devise the Likelihood Ratio Test (LRT) for cutoff :

The Significance of the LRT is in that it is the most powerful test:

thm. Neymann—Pearson Lemma. Given level [=false positive rate = type I error rate], then the LRT is the Uniformly Most Powerful (UMP) test.

There is also another significant result for certain likelihood ratio. But first define:

def. Monotone Likelihood Ratio (MLR) is when the likelihood ratio is never decreasing for some variable .

Then we can determine the UMP of such :

thm. Karlin—Rubin Theorem. In a distribution with that is MLR for region [statistic] for a sampling in this distribution the UMP is: