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)