Uniformly most powerful test

In statistical hypothesis testing, a uniformly most powerful (UMP) test is a hypothesis test which has the greatest power among all possible tests of a given size α. For example, according to the Neyman–Pearson lemma, the likelihood-ratio test is UMP for testing simple (point) hypotheses. Let denote a random vector (corresponding to the measurements), taken from a parametrized family of probability density functions or probability mass functions , which depends on the unknown deterministic parameter . The parameter space is partitioned into two disjoint sets and . Let denote the hypothesis that , and let denote the hypothesis that . The binary test of hypotheses is performed using a test function with a reject region (a subset of measurement space). meaning that is in force if the measurement and that is in force if the measurement . Note that is a disjoint covering of the measurement space. A test function is UMP of size if for any other test function 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 . If is monotone non-decreasing, in , for any pair (meaning that the greater is, the more likely is), then the threshold test: where is chosen such that is the UMP test of size α for testing Note that exactly the same test is also UMP for testing Although the Karlin-Rubin theorem may seem weak because of its restriction to scalar parameter and scalar measurement, it turns out that there exist a host of problems for which the theorem holds. In particular, the one-dimensional exponential family of probability density functions or probability mass functions with has a monotone non-decreasing likelihood ratio in the sufficient statistic , provided that is non-decreasing. Let denote i.i.d. normally distributed -dimensional random vectors with mean and covariance matrix .