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A function characterizing the quality of a [[Statistical test|statistical test]]. Suppose that, based on a realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p0742101.png" /> of a random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p0742102.png" /> with values in a sampling space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p0742103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p0742104.png" />, it is necessary to test the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p0742105.png" /> according to which the probability distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p0742106.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p0742107.png" /> belongs to a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p0742108.png" />, against the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p0742109.png" /> according to which
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421010.png" /></td> </tr></table>
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and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421011.png" /> be the critical function of the statistical test intended for testing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421012.png" /> against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421013.png" />. Then
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$
 +
{\mathsf P} _  \theta  \in  H _ {1}  = \
 +
\{ { {\mathsf P} _  \theta  } : {\theta \in \Theta _ {1} = \Theta \setminus
 +
\Theta _ {0} } \}
 +
,
 +
$$
  
is called the power function of the statistical test with critical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421015.png" />. It follows from (*) that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421016.png" /> gives the probabilities with which the statistical test for testing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421017.png" /> against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421018.png" /> rejects the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421019.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421020.png" /> is subject to the law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421022.png" />.
+
and let  $  \phi ( \cdot ) $
 +
be the critical function of the statistical test intended for testing $  H _ {0} $
 +
against $  H _ {1} $.  
 +
Then
  
In the theory of statistical hypothesis testing, founded by J. Neyman and E. Pearson, the problem of testing a compound hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421023.png" /> against a compound alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421024.png" /> is formulated in terms of the power function of a test and consists of the construction of a test maximizing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421025.png" />, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421026.png" />, under the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421028.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421029.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421030.png" />) is called the significance level of the test — a given admissible probability of the error of rejecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074210/p07421031.png" /> when it is in fact true.
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$$ \tag{* }
 +
\beta ( \theta )  = \
 +
\int\limits _ { \mathfrak X } \phi ( x)  d {\mathsf P} _  \theta  ( x) ,\ \
 +
\theta \in \Theta = \Theta _ {0} \cup \Theta _ {1} ,
 +
$$
 +
 
 +
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====
 
====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>
 
<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>

Latest revision as of 08:07, 6 June 2020


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=14564
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article