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A region of a sample space with the property that if the observed value of a random variable, the distribution of which is connected with the tested hypothesis, falls in that region, then the hypothesis is rejected. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c0271001.png" /> be the tested hypothesis concerning the distribution of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c0271002.png" />, taking values in a sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c0271003.png" />. In constructing a non-randomized test for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c0271004.png" />, one divides the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c0271005.png" /> into two disjoint sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c0271006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c0271007.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c0271008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c0271009.png" />. The test amounts to the following rule: Reject <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c02710010.png" /> if an experimental realization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c02710011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c02710012.png" /> falls in the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c02710013.png" />; accept <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c02710014.png" /> otherwise (i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c02710015.png" />). The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c02710016.png" /> is called the critical region of the test; its complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c02710017.png" /> is called the acceptance region. In this sense, the problem of selecting a critical region is equivalent to the construction of a non-randomized statistical test for the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c02710018.png" />. Naturally, the critical region is decided upon before sampling to test for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027100/c02710019.png" />; on the other hand, within the context of the Neyman–Pearson theory, the actual choice of the critical region is determined by the probabilities of the errors of the first and second kind occurring in problems of statistical hypotheses testing.
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A region of a [[sample space]] with the property that if the observed value of a [[random variable]], the distribution of which is connected with the tested hypothesis, falls in that region, then the hypothesis is rejected (cf. [[Statistical hypotheses, verification of]]). Let $  H _ {0} $
 +
be the tested hypothesis concerning the distribution of a random variable $  X $,  
 +
taking values in a sample space $  ( \mathfrak X , \mathfrak B ) $.  
 +
In constructing a non-randomized test for $  H _ {0} $,  
 +
one divides the space $  \mathfrak X $
 +
into two disjoint sets $  K $
 +
and $  \overline{K}\; $
 +
such that $  K \cup \overline{K}\; = \mathfrak X $,  
 +
$  K \in \mathfrak B $.  
 +
The test amounts to the following rule: Reject $  H _ {0} $
 +
if an experimental realization $  x $
 +
of $  X $
 +
falls in the set $  K $;  
 +
accept $  H _ {0} $
 +
otherwise (i.e. if $  x \in \overline{K}\;  $).  
 +
The set $  K $
 +
is called the critical region of the test; its complement $  \overline{K}\; $
 +
is called the acceptance region. In this sense, the problem of selecting a critical region is equivalent to the construction of a non-randomized statistical test for the hypothesis $  H _ {0} $.  
 +
Naturally, the critical region is decided upon before sampling to test for $  H _ {0} $;  
 +
on the other hand, within the context of the Neyman–Pearson theory, the actual choice of the critical region is determined by the probabilities of the errors of the first and second kind occurring in problems of statistical hypotheses testing (cf. [[Neyman–Pearson lemma]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1959)</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">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1959)</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 19:26, 30 December 2020


A region of a sample space with the property that if the observed value of a random variable, the distribution of which is connected with the tested hypothesis, falls in that region, then the hypothesis is rejected (cf. Statistical hypotheses, verification of). Let $ H _ {0} $ be the tested hypothesis concerning the distribution of a random variable $ X $, taking values in a sample space $ ( \mathfrak X , \mathfrak B ) $. In constructing a non-randomized test for $ H _ {0} $, one divides the space $ \mathfrak X $ into two disjoint sets $ K $ and $ \overline{K}\; $ such that $ K \cup \overline{K}\; = \mathfrak X $, $ K \in \mathfrak B $. The test amounts to the following rule: Reject $ H _ {0} $ if an experimental realization $ x $ of $ X $ falls in the set $ K $; accept $ H _ {0} $ otherwise (i.e. if $ x \in \overline{K}\; $). The set $ K $ is called the critical region of the test; its complement $ \overline{K}\; $ is called the acceptance region. In this sense, the problem of selecting a critical region is equivalent to the construction of a non-randomized statistical test for the hypothesis $ H _ {0} $. Naturally, the critical region is decided upon before sampling to test for $ H _ {0} $; on the other hand, within the context of the Neyman–Pearson theory, the actual choice of the critical region is determined by the probabilities of the errors of the first and second kind occurring in problems of statistical hypotheses testing (cf. Neyman–Pearson lemma).

References

[1] H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[2] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959)
[3] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)
How to Cite This Entry:
Critical region. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Critical_region&oldid=11738
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article