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''similar region''
 
''similar region''
  
 
A generally used abbreviation of the term  "critical region similar to a sample space"  as used in mathematical statistics for a [[Critical region|critical region]] with non-randomized similarity of a statistical test.
 
A generally used abbreviation of the term  "critical region similar to a sample space"  as used in mathematical statistics for a [[Critical region|critical region]] with non-randomized similarity of a statistical test.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851701.png" /> be a random variable taking values in a sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851702.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851703.png" />, and consider testing the compound hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851704.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851705.png" /> against the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851706.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851707.png" />. Suppose that in order to test <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851708.png" /> against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s0851709.png" />, a non-randomized [[Similar test|similar test]] of level <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517010.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517011.png" />) has been constructed, with critical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517013.png" />. As this test is non-randomized,
+
Let $  X $
 +
be a random variable taking values in a sample space $  ( \mathfrak X , \mathfrak B , {\mathsf P} _  \theta  ) $,  
 +
$  \theta \in \Theta $,  
 +
and consider testing the compound hypothesis $  H _ {0} $:  
 +
$  \theta \in \Theta _ {0} \subset  \Theta $
 +
against the alternative $  H _ {1} $:  
 +
$  \theta \in \Theta _ {1} = \Theta \setminus  \Theta _ {0} $.  
 +
Suppose that in order to test $  H _ {0} $
 +
against $  H _ {1} $,  
 +
a non-randomized [[Similar test|similar test]] of level $  \alpha $(
 +
$  0 < \alpha < 1 $)  
 +
has been constructed, with critical function $  \phi ( x) $,  
 +
$  x \in \mathfrak X $.  
 +
As this test is non-randomized,
 +
 
 +
$$ \tag{1 }
 +
\phi ( x)  = \
 +
\left \{
  
<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/s/s085/s085170/s08517014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\begin{array}{ll}
 +
1,  &{ x \in K \subset  \mathfrak X, }  \\
 +
0,  &{ x \notin K, }  \\
 +
\end{array}
 +
\right .
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517015.png" /> is a certain set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517016.png" />, called the critical set for the test (according to this test, the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517017.png" /> is rejected in favour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517018.png" /> if the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517019.png" /> is observed in an experiment). Also, the constructed test is a similar test, which means that
+
where $  K $
 +
is a certain set in $  \mathfrak X $,  
 +
called the critical set for the test (according to this test, the hypothesis $  H _ {0} $
 +
is rejected in favour of $  H _ {1} $
 +
if the event $  \{ X \in K \} $
 +
is observed in an experiment). Also, the constructed test is a similar test, which means that
  
<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/s/s085/s085170/s08517020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\int\limits _ { \mathfrak X } \phi ( x)  d {\mathsf P} _  \theta  = \alpha
 +
\  \textrm{ for }  \textrm{ all }  \theta \in \Theta _ {0} .
 +
$$
  
It follows from (1) and (2) that the critical region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517021.png" /> of a non-randomized similar test has the property:
+
It follows from (1) and (2) that the critical region $  K $
 +
of a non-randomized similar test has the property:
  
<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/s/s085/s085170/s08517022.png" /></td> </tr></table>
+
$$
 +
{\mathsf P} _  \theta  \{ X \in K \}  = \alpha
 +
\  \textrm{ for }  \textrm{ all }  \theta \in \Theta _ {0} .
 +
$$
  
Accordingly, J. Neyman and E.S. Pearson emphasized the latter feature of the critical set of a non-randomized similar test and called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517023.png" /> a  "region similar to the sample space"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517024.png" />, in the sense that the two probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517026.png" /> are independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085170/s08517027.png" />.
+
Accordingly, J. Neyman and E.S. Pearson emphasized the latter feature of the critical set of a non-randomized similar test and called $  K $
 +
a  "region similar to the sample space"   $ \mathfrak X $,  
 +
in the sense that the two probabilities $  {\mathsf P} _  \theta  \{ X \in K \} $
 +
and $  {\mathsf P} _  \theta  \{ X \in \mathfrak X \} $
 +
are independent of $  \theta \in \Theta _ {0} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. van der Waerden,  "Mathematische Statistik" , Springer  (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Neyman,  E.S. Pearson,  "On the problem of the most efficient tests of statistical hypotheses"  ''Philos. Trans. Roy. Soc. London Ser. A'' , '''231'''  (1933)  pp. 289–337</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.L. Lehmann,  H. Scheffé,  "Completeness, similar regions, and unbiased estimation I"  ''Sankhyā'' , '''10'''  (1950)  pp. 305–340</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.L. Lehmann,  H. Scheffé,  "Completeness, similar regions, and unbiased estimation II"  ''Sankhyā'' , '''15'''  (1955)  pp. 219–236</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.L. van der Waerden,  "Mathematische Statistik" , Springer  (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Neyman,  E.S. Pearson,  "On the problem of the most efficient tests of statistical hypotheses"  ''Philos. Trans. Roy. Soc. London Ser. A'' , '''231'''  (1933)  pp. 289–337</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.L. Lehmann,  H. Scheffé,  "Completeness, similar regions, and unbiased estimation I"  ''Sankhyā'' , '''10'''  (1950)  pp. 305–340</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.L. Lehmann,  H. Scheffé,  "Completeness, similar regions, and unbiased estimation II"  ''Sankhyā'' , '''15'''  (1955)  pp. 219–236</TD></TR></table>

Latest revision as of 14:55, 7 June 2020


similar region

A generally used abbreviation of the term "critical region similar to a sample space" as used in mathematical statistics for a critical region with non-randomized similarity of a statistical test.

Let $ X $ be a random variable taking values in a sample space $ ( \mathfrak X , \mathfrak B , {\mathsf P} _ \theta ) $, $ \theta \in \Theta $, and consider testing the compound hypothesis $ H _ {0} $: $ \theta \in \Theta _ {0} \subset \Theta $ against the alternative $ H _ {1} $: $ \theta \in \Theta _ {1} = \Theta \setminus \Theta _ {0} $. Suppose that in order to test $ H _ {0} $ against $ H _ {1} $, a non-randomized similar test of level $ \alpha $( $ 0 < \alpha < 1 $) has been constructed, with critical function $ \phi ( x) $, $ x \in \mathfrak X $. As this test is non-randomized,

$$ \tag{1 } \phi ( x) = \ \left \{ \begin{array}{ll} 1, &{ x \in K \subset \mathfrak X, } \\ 0, &{ x \notin K, } \\ \end{array} \right . $$

where $ K $ is a certain set in $ \mathfrak X $, called the critical set for the test (according to this test, the hypothesis $ H _ {0} $ is rejected in favour of $ H _ {1} $ if the event $ \{ X \in K \} $ is observed in an experiment). Also, the constructed test is a similar test, which means that

$$ \tag{2 } \int\limits _ { \mathfrak X } \phi ( x) d {\mathsf P} _ \theta = \alpha \ \textrm{ for } \textrm{ all } \theta \in \Theta _ {0} . $$

It follows from (1) and (2) that the critical region $ K $ of a non-randomized similar test has the property:

$$ {\mathsf P} _ \theta \{ X \in K \} = \alpha \ \textrm{ for } \textrm{ all } \theta \in \Theta _ {0} . $$

Accordingly, J. Neyman and E.S. Pearson emphasized the latter feature of the critical set of a non-randomized similar test and called $ K $ a "region similar to the sample space" $ \mathfrak X $, in the sense that the two probabilities $ {\mathsf P} _ \theta \{ X \in K \} $ and $ {\mathsf P} _ \theta \{ X \in \mathfrak X \} $ are independent of $ \theta \in \Theta _ {0} $.

References

[1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[2] B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)
[3] J. Neyman, E.S. Pearson, "On the problem of the most efficient tests of statistical hypotheses" Philos. Trans. Roy. Soc. London Ser. A , 231 (1933) pp. 289–337
[4] E.L. Lehmann, H. Scheffé, "Completeness, similar regions, and unbiased estimation I" Sankhyā , 10 (1950) pp. 305–340
[5] E.L. Lehmann, H. Scheffé, "Completeness, similar regions, and unbiased estimation II" Sankhyā , 15 (1955) pp. 219–236
How to Cite This Entry:
Similarity region. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Similarity_region&oldid=49424
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article