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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068170/o0681702.png" />-distribution''
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'' $  \omega  ^ {2} $-
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distribution''
  
 
The [[Probability distribution|probability distribution]] of the random variable
 
The [[Probability distribution|probability distribution]] of the random variable
  
<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/o/o068/o068170/o0681703.png" /></td> </tr></table>
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$$
 +
\omega  ^ {2}  = \int\limits _ { 0 } ^ { 1 }  Z  ^ {2} ( t)  dt,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068170/o0681704.png" /> is a conditional [[Wiener process|Wiener process]] (conditioned on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068170/o0681705.png" />). The characteristic function of the  "omega-squared"  distribution is expressed by the formula
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where $  Z( t) $
 +
is a conditional [[Wiener process|Wiener process]] (conditioned on $  Z ( 1) = 0 $).  
 +
The characteristic function of the  "omega-squared"  distribution is expressed by the formula
  
<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/o/o068/o068170/o0681706.png" /></td> </tr></table>
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$$
 +
{\mathsf E} e ^ {it \omega  ^ {2} }  = \prod _ { k= 1} ^  \infty 
 +
\left ( 1 -
 +
\frac{2it }{\pi  ^ {2} k  ^ {2} }
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\right )  ^ {-1/2} .
 +
$$
  
In mathematical statistics, the  "omega-squared"  distribution is often found in the following circumstances. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068170/o0681707.png" /> be independent random variables, uniformly distributed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068170/o0681708.png" />, according to which an empirical distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068170/o0681709.png" /> is constructed. In this case, the process
+
In mathematical statistics, the  "omega-squared"  distribution is often found in the following circumstances. Let $  X _ {1} \dots X _ {n} $
 +
be independent random variables, uniformly distributed on $  [ 0, 1] $,
 +
according to which an empirical distribution function $  F _ {n} ( \cdot ) $
 +
is constructed. In this case, the process
  
<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/o/o068/o068170/o06817010.png" /></td> </tr></table>
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$$
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Z _ {n} ( t)  = \sqrt n ( F _ {n} ( t) - t)
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$$
  
 
converges weakly to a conditional Wiener process, from which it follows that
 
converges weakly to a conditional Wiener process, from which it follows 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/o/o068/o068170/o06817011.png" /></td> </tr></table>
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$$
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\lim\limits _ {n \rightarrow \infty }  {\mathsf P} \left \{ \int\limits _ { 0 } ^ { 1 }  Z _ {n}  ^ {2} ( t)  dt < \lambda \right \}  = \
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{\mathsf P} \{ \omega  ^ {2} < \lambda \} =
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$$
  
<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/o/o068/o068170/o06817012.png" /></td> </tr></table>
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$$
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= \
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1 -
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\frac{2} \pi
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\sum _ { k= 1} ^  \infty  (- 1)  ^ {k-1} \int\limits _ {( 2k- 1) \pi } ^ { {2k }  \pi }
 +
\frac{e ^ {- t  ^ {2} \lambda / 2 }
 +
}{\sqrt {- t  \sin  t } }
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  dt,\  \lambda > 0 .
 +
$$
  
 
See also [[Cramér–von Mises test|Cramér–von Mises test]].
 
See also [[Cramér–von Mises test|Cramér–von Mises test]].
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.V. Smirnov,  "On the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068170/o06817013.png" />-distribution"  ''Mat. Sb.'' , '''2'''  (1937)  pp. 973–993  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.W. Anderson,  D.A. Darling,  "Asymptotic theory of certain  "goodness of fit"  criteria based on stochastic processes"  ''Ann. Math. Stat.'' , '''23'''  (1952)  pp. 193–212</TD></TR></table>
 
 
 
  
 
====Comments====
 
====Comments====
The  "conditional Wiener process"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068170/o06817014.png" /> is usually referred to in the Western literature as tied-down Brownian motion, pinned Brownian motion or as the Brownian bridge.
+
The  "conditional Wiener process"   $ Z $
 +
is usually referred to in the Western literature as tied-down Brownian motion, pinned Brownian motion or as the Brownian bridge.
  
 
The pioneering paper is [[#References|[a1]]].
 
The pioneering paper is [[#References|[a1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.A. Darling,  "The Cramér–Smirnov test in the parametric case"  ''Ann. Math. Stat.'' , '''26'''  (1955)  pp. 1–20</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Durbin,  "Distribution theory for tests based on the sample distribution function" , SIAM  (1973)</TD></TR></table>
+
<table><tr><td valign="top">[1]</td> <td valign="top">  N.V. Smirnov,  "On the $\omega ^ { 2 }$-distribution"  ''Mat. Sb.'' , '''2'''  (1937)  pp. 973–993  (In Russian)</td></tr><tr><td valign="top">[2]</td> <td valign="top">  T.W. Anderson,  D.A. Darling,  "Asymptotic theory of certain  "goodness of fit"  criteria based on stochastic processes"  ''Ann. Math. Stat.'' , '''23'''  (1952)  pp. 193–212</td></tr>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  D.A. Darling,  "The Cramér–Smirnov test in the parametric case"  ''Ann. Math. Stat.'' , '''26'''  (1955)  pp. 1–20</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Durbin,  "Distribution theory for tests based on the sample distribution function" , SIAM  (1973)</td></tr></table>

Latest revision as of 20:47, 22 January 2024


$ \omega ^ {2} $- distribution

The probability distribution of the random variable

$$ \omega ^ {2} = \int\limits _ { 0 } ^ { 1 } Z ^ {2} ( t) dt, $$

where $ Z( t) $ is a conditional Wiener process (conditioned on $ Z ( 1) = 0 $). The characteristic function of the "omega-squared" distribution is expressed by the formula

$$ {\mathsf E} e ^ {it \omega ^ {2} } = \prod _ { k= 1} ^ \infty \left ( 1 - \frac{2it }{\pi ^ {2} k ^ {2} } \right ) ^ {-1/2} . $$

In mathematical statistics, the "omega-squared" distribution is often found in the following circumstances. Let $ X _ {1} \dots X _ {n} $ be independent random variables, uniformly distributed on $ [ 0, 1] $, according to which an empirical distribution function $ F _ {n} ( \cdot ) $ is constructed. In this case, the process

$$ Z _ {n} ( t) = \sqrt n ( F _ {n} ( t) - t) $$

converges weakly to a conditional Wiener process, from which it follows that

$$ \lim\limits _ {n \rightarrow \infty } {\mathsf P} \left \{ \int\limits _ { 0 } ^ { 1 } Z _ {n} ^ {2} ( t) dt < \lambda \right \} = \ {\mathsf P} \{ \omega ^ {2} < \lambda \} = $$

$$ = \ 1 - \frac{2} \pi \sum _ { k= 1} ^ \infty (- 1) ^ {k-1} \int\limits _ {( 2k- 1) \pi } ^ { {2k } \pi } \frac{e ^ {- t ^ {2} \lambda / 2 } }{\sqrt {- t \sin t } } dt,\ \lambda > 0 . $$

See also Cramér–von Mises test.

Comments

The "conditional Wiener process" $ Z $ is usually referred to in the Western literature as tied-down Brownian motion, pinned Brownian motion or as the Brownian bridge.

The pioneering paper is [a1].

References

[1] N.V. Smirnov, "On the $\omega ^ { 2 }$-distribution" Mat. Sb. , 2 (1937) pp. 973–993 (In Russian)
[2] T.W. Anderson, D.A. Darling, "Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes" Ann. Math. Stat. , 23 (1952) pp. 193–212
[a1] D.A. Darling, "The Cramér–Smirnov test in the parametric case" Ann. Math. Stat. , 26 (1955) pp. 1–20
[a2] J. Durbin, "Distribution theory for tests based on the sample distribution function" , SIAM (1973)
How to Cite This Entry:
Omega-squared distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Omega-squared_distribution&oldid=17313
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article