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A function characterizing the quality of a [[Statistical test|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Cramér,   "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,   "Mathematische Statistik" , Springer  (1957)</TD></TR></table>