Power function of a test

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A function characterizing the quality of a statistical test. Suppose that, based on a realization of a random vector with values in a sampling space , , it is necessary to test the hypothesis according to which the probability distribution of belongs to a subset , against the alternative according to which

and let be the critical function of the statistical test intended for testing against . Then


is called the power function of the statistical test with critical function . It follows from (*) that gives the probabilities with which the statistical test for testing against rejects the hypothesis if is subject to the law , .

In the theory of statistical hypothesis testing, founded by J. Neyman and E. Pearson, the problem of testing a compound hypothesis against a compound alternative is formulated in terms of the power function of a test and consists of the construction of a test maximizing , when , under the condition that for all , where () is called the significance level of the test — a given admissible probability of the error of rejecting when it is in fact true.


[1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959)
[2] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[3] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)
How to Cite This Entry:
Power function of a test. M.S. Nikulin (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098