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A statistic for which the values are the conditional probabilities of the deviations from the hypothesis being tested, given the value of an observed result. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c0270601.png" /> be a random variable with values in a sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c0270602.png" />, the distribution of which belongs to a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c0270603.png" />, and suppose one is testing the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c0270604.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c0270605.png" />, against the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c0270606.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c0270607.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c0270608.png" /> be a measurable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c0270609.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c02706010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c02706011.png" />. If the hypothesis is being tested by a randomized test, according to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c02706012.png" /> is rejected with probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c02706013.png" /> if the experiment reveals that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c02706014.png" />, and accepted with probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c02706015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c02706016.png" /> is called the critical function of the test. In setting up a non-randomized test, one chooses the critical function in such a way that it assumes only two values, 0 and 1. Hence it is the characteristic function of a certain set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c02706017.png" />, called the critical region of the test: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c02706018.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c02706019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c02706020.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027060/c02706021.png" />.
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A statistic for which the values are the conditional probabilities of the deviations from the hypothesis being tested, given the value of an observed result. Let $  X $
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be a random variable with values in a sample space $  ( \mathfrak X , \mathfrak B ) $,  
 +
the distribution of which belongs to a family $  \{ {P _  \theta  } : {\theta \in \Theta } \} $,  
 +
and suppose one is testing the hypothesis $  H _ {0} $:  
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$  \theta \in \Theta _ {0} \subset  \Theta $,  
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against the alternative $  H _ {1} $:  
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$  \theta \in \Theta _ {1} = \Theta \setminus  \Theta _ {0} $.  
 +
Let $  \phi ( \cdot ) $
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be a measurable function on $  \mathfrak X $
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such that 0 \leq  \phi ( x) \leq  1 $
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for all $  x \in \mathfrak X $.  
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If the hypothesis is being tested by a randomized test, according to which $  H _ {0} $
 +
is rejected with probability $  \phi ( x) $
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if the experiment reveals that $  X = x $,  
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and accepted with probability $  1 - \phi ( x) $,  
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then $  \phi ( \cdot ) $
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is called the critical function of the test. In setting up a non-randomized test, one chooses the critical function in such a way that it assumes only two values, 0 and 1. Hence it is the characteristic function of a certain set $  K \in \mathfrak B $,  
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called the critical region of the test: $  \phi ( x) = 1 $
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if $  x \in K $,  
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$  \phi ( x) = 0 $
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if $  x \notin K $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1959)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1959)</TD></TR></table>

Latest revision as of 17:31, 5 June 2020


A statistic for which the values are the conditional probabilities of the deviations from the hypothesis being tested, given the value of an observed result. Let $ X $ be a random variable with values in a sample space $ ( \mathfrak X , \mathfrak B ) $, the distribution of which belongs to a family $ \{ {P _ \theta } : {\theta \in \Theta } \} $, and suppose one is testing the hypothesis $ H _ {0} $: $ \theta \in \Theta _ {0} \subset \Theta $, against the alternative $ H _ {1} $: $ \theta \in \Theta _ {1} = \Theta \setminus \Theta _ {0} $. Let $ \phi ( \cdot ) $ be a measurable function on $ \mathfrak X $ such that $ 0 \leq \phi ( x) \leq 1 $ for all $ x \in \mathfrak X $. If the hypothesis is being tested by a randomized test, according to which $ H _ {0} $ is rejected with probability $ \phi ( x) $ if the experiment reveals that $ X = x $, and accepted with probability $ 1 - \phi ( x) $, then $ \phi ( \cdot ) $ is called the critical function of the test. In setting up a non-randomized test, one chooses the critical function in such a way that it assumes only two values, 0 and 1. Hence it is the characteristic function of a certain set $ K \in \mathfrak B $, called the critical region of the test: $ \phi ( x) = 1 $ if $ x \in K $, $ \phi ( x) = 0 $ if $ x \notin K $.

References

[1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959)
How to Cite This Entry:
Critical function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Critical_function&oldid=13465
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article