Letdenote a random vector (corresponding to the measurements), taken from aparametrized familyofprobability density functionsorprobability mass functions,which depends on the unknown deterministic parameter.The parameter spaceis partitioned into two disjoint setsand.Letdenote the hypothesis that,and letdenote the hypothesis that.
The binary test of hypotheses is performed using a test functionwith a reject region(a subset of measurement space).
meaning thatis in force if the measurementand thatis in force if the measurement.
Note thatis a disjoint covering of the measurement space.
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 parameterθ,and define the likelihood ratio.
Ifis monotone non-decreasing, in,for any pair(meaning that the greateris, the more likelyis), then the threshold test:
whereis chosen such that
is the UMP test of sizeαfor 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-dimensionalexponential familyofprobability density functionsorprobability mass functionswith
has a monotone non-decreasing likelihood ratio in thesufficient statistic,provided thatis non-decreasing.
Finally, we note that 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. forwhere) is different from the most powerful test of the same size for a different value of the parameter (e.g. forwhere). As a result, no test isuniformlymost powerful in these situations.
Ferguson, T. S.(1967). "Sec. 5.2:Uniformly most powerful tests".Mathematical Statistics: A decision theoretic approach.New York: Academic Press.
Mood, A. M.; Graybill, F. A.; Boes, D. C. (1974). "Sec. IX.3.2:Uniformly most powerful tests".Introduction to the theory of statistics(3rd ed.). New York: McGraw-Hill.
L. L. Scharf,Statistical Signal Processing,Addison-Wesley, 1991, section 4.7.