# Power function of a test

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.

#### References

[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: http://www.encyclopediaofmath.org/index.php?title=Power_function_of_a_test&oldid=14564