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)