# Power function of a test

A function characterizing the quality of a statistical test. Suppose that, based on a realization $ x $
of a random vector $ X $
with values in a sampling space $ ( X , B , {\mathsf P} _ \theta ) $,
$ \theta \in \Theta $,
it is necessary to test the hypothesis $ H _ {0} $
according to which the probability distribution $ {\mathsf P} _ \theta $
of $ X $
belongs to a subset $ H _ {0} = \{ { {\mathsf P} _ \theta } : {\theta \in \Theta _ {0} \subset \Theta } \} $,
against the alternative $ H _ {1} $
according to which

$$ {\mathsf P} _ \theta \in H _ {1} = \ \{ { {\mathsf P} _ \theta } : {\theta \in \Theta _ {1} = \Theta \setminus \Theta _ {0} } \} , $$

and let $ \phi ( \cdot ) $ be the critical function of the statistical test intended for testing $ H _ {0} $ against $ H _ {1} $. Then

$$ \tag{* } \beta ( \theta ) = \ \int\limits _ { \mathfrak X } \phi ( x) d {\mathsf P} _ \theta ( x) ,\ \ \theta \in \Theta = \Theta _ {0} \cup \Theta _ {1} , $$

is called the power function of the statistical test with critical function $ \phi $. It follows from (*) that $ \beta ( \theta ) $ gives the probabilities with which the statistical test for testing $ H _ {0} $ against $ H _ {1} $ rejects the hypothesis $ H _ {0} $ if $ X $ is subject to the law $ {\mathsf P} _ \theta $, $ \theta \in \Theta $.

In the theory of statistical hypothesis testing, founded by J. Neyman and E. Pearson, the problem of testing a compound hypothesis $ H _ {0} $ against a compound alternative $ H _ {1} $ is formulated in terms of the power function of a test and consists of the construction of a test maximizing $ \beta ( \theta ) $, when $ \theta \in \Theta $, under the condition that $ \beta ( \theta ) \leq \alpha $ for all $ \theta \in \Theta _ {0} $, where $ \alpha $( $ 0 < \alpha < 1 $) is called the significance level of the test — a given admissible probability of the error of rejecting $ H _ {0} $ 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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Power_function_of_a_test&oldid=48272