 A test function formula_12 is UMP of size formula_20 if for any other test function formula_21 satisfying
we have

The Karlin–Rubin theorem can be regarded as an extension of the Neyman–Pearson lemma for composite hypotheses.Consider a scalar measurement having a probability density function parameterized by a scalar parameter "θ", and define the likelihood ratio formula_24.If formula_25 is monotone non-decreasing, in formula_26, for any pair formula_27 (meaning that the greater formula_26 is, the more likely formula_10 is), then the threshold test