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.
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
denote the hypothesis that
denote the hypothesis that
The binary test of hypotheses is performed using a test function
with a reject region
(a subset of measurement space).
is in force if the measurement
is in force if the measurement
is a disjoint covering of the measurement space.
is UMP of size
satisfying we have The Karlin–Rubin theorem can be regarded as an extension of the Neyman–Pearson lemma for composite hypotheses.
[1] Consider a scalar measurement having a probability density function parameterized by a scalar parameter θ, and define the likelihood ratio
is monotone non-decreasing, in
(meaning that the greater
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
η ( θ )
-dimensional random vectors with mean
and covariance matrix
We then have which is exactly in the form of the exponential family shown in the previous section, with the sufficient statistic being Thus, we conclude that the test is the UMP test of size
In general, UMP tests do not exist for vector parameters or for two-sided tests (a test in which one hypothesis lies on both sides of the alternative).
The reason is that in these situations, the most powerful test of a given size for one possible value of the parameter (e.g. for
) is different from the most powerful test of the same size for a different value of the parameter (e.g. for
As a result, no test is uniformly most powerful in these situations.