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Difference between revisions of "Similar statistic"

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$$ \tag{* }
 
$$ \tag{* }
{\mathsf P} _  \theta  ( T  ^ {-} 1 ( B)) \  \textrm{ is }  \textrm{ independent } \  
+
{\mathsf P} _  \theta  ( T  ^ {- 1} ( B)) \  \textrm{ is }  \textrm{ independent } \  
 
\textrm{ of }  \theta  \textrm{ for }  \theta \in \Theta _ {0} ,
 
\textrm{ of }  \theta  \textrm{ for }  \theta \in \Theta _ {0} ,
 
$$
 
$$
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$$  
 
$$  
T  =  \left ( {\sum _ { i= } 1 ^ { n }  ( X _ {i} - \overline{X}\; )  ^ {2} } \right ) ^  
+
T  =  \left ( {\sum _ { i=1} ^ { n }  ( X _ {i} - \overline{X}\; )  ^ {2} } \right ) ^  
{- \alpha } \sum _ { i= } 1 ^ { n }  ( X _ {i} - \overline{X}\; ) ^ {2 \alpha } ,
+
{- \alpha } \sum _ { i=1} ^ { n }  ( X _ {i} - \overline{X}\; ) ^ {2 \alpha } ,
 
$$
 
$$
  
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\overline{X}\;  =   
 
\overline{X}\;  =   
 
\frac{1}{n}
 
\frac{1}{n}
  \sum _ { i= } 1 ^ { n }  X _ {i} ,
+
  \sum _ { i=1} ^ { n }  X _ {i} ,
 
$$
 
$$
  
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.-L. Soler,  "Basic structures in mathematical statistics" , Moscow  (1972)  (In Russian; translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.V. Linnik,  "Statistical problems with nuisance parameters" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-R. Barra,  "Mathematical bases of statistics" , Acad. Press  (1981)  (Translated from French)</TD></TR></table>
+
<table>
 
+
<TR><TD valign="top">[1]</TD> <TD valign="top">  J.-L. Soler,  "Basic structures in mathematical statistics" , Moscow  (1972)  (In Russian; translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Yu.V. Linnik,  "Statistical problems with nuisance parameters" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.-R. Barra,  "Mathematical bases of statistics" , Acad. Press  (1981)  (Translated from French)</TD></TR>
====Comments====
+
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR>
 
+
</table>
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1986)</TD></TR></table>
 

Revision as of 08:53, 6 January 2024


A statistic having a fixed probability distribution under some compound hypothesis.

Let the statistic $ T $ map the sample space $ ( \mathfrak X , {\mathcal B} _ {\mathfrak X} , {\mathsf P} _ \theta ) $, $ \theta \in \Theta $, into a measurable space $ ( \mathfrak A, {\mathcal B} _ {\mathfrak Y} ) $ and consider some compound hypothesis $ H _ {0} $: $ \theta \in \Theta _ {0} \subseteq \Theta $. In that case, if for any event $ B \in {\mathcal B} _ {\mathfrak Y} $ the probability

$$ \tag{* } {\mathsf P} _ \theta ( T ^ {- 1} ( B)) \ \textrm{ is } \textrm{ independent } \ \textrm{ of } \theta \textrm{ for } \theta \in \Theta _ {0} , $$

one says that $ T $ is a similar statistic with respect to $ H _ {0} $, or simply that it is a similar statistic. It is clear that condition (*) is equivalent to saying that the distribution of the statistic $ T $ does not vary when $ \theta $ runs through $ \Theta _ {0} $. With this property in view, it is frequently said of a similar statistic that it is independent of the parameter $ \theta $, $ \theta \in \Theta _ {0} $. Similar statistics play a large role in constructing similar tests, and also in solving statistical problems with nuisance parameters.

Example 1. Let $ X _ {1} \dots X _ {n} $ be independent random variables with identical normal distribution $ N _ {1} ( a, \sigma ^ {2} ) $ with $ | a | < \infty $ and $ \sigma > 0 $. Then for any $ \alpha > 0 $ the statistic

$$ T = \left ( {\sum _ { i=1} ^ { n } ( X _ {i} - \overline{X}\; ) ^ {2} } \right ) ^ {- \alpha } \sum _ { i=1} ^ { n } ( X _ {i} - \overline{X}\; ) ^ {2 \alpha } , $$

where

$$ \overline{X}\; = \frac{1}{n} \sum _ { i=1} ^ { n } X _ {i} , $$

is independent of the two-dimensional parameter $ ( a, \sigma ^ {2} ) $.

Example 2. Let $ X _ {1} \dots X _ {n+} m $ be independent identically-distributed random variables whose distribution functions belong to the family $ {\mathcal F} = \{ F ( x) \} $ of all continuous distribution functions on $ ( - \infty , + \infty ) $. If $ F _ {n} ( x) $ and $ F _ {m} ( x) $ are empirical distribution functions constructed from the observations $ X _ {1} \dots X _ {n} $ and $ X _ {n+} 1 \dots X _ {n+} m $, respectively, then the Smirnov statistic

$$ S _ {n,m} = \sup _ {| x| < \infty } | F _ {n} ( x) - F _ {m} ( x) | $$

is similar with respect to the family $ {\mathcal F} $.

References

[1] J.-L. Soler, "Basic structures in mathematical statistics" , Moscow (1972) (In Russian; translated from French)
[2] Yu.V. Linnik, "Statistical problems with nuisance parameters" , Amer. Math. Soc. (1968) (Translated from Russian)
[3] J.-R. Barra, "Mathematical bases of statistics" , Acad. Press (1981) (Translated from French)
[a1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
How to Cite This Entry:
Similar statistic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Similar_statistic&oldid=54844
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article