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The joint distribution of the elements from the sample covariance matrix of observations from a multivariate [[Normal distribution|normal distribution]]. Let the results of observations have a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w0980401.png" />-dimensional normal distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w0980402.png" /> with vector mean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w0980403.png" /> and covariance matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w0980404.png" />. Then the joint density of the elements of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w0980405.png" /> is given by the formula
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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/w/w098/w098040/w0980406.png" /></td> </tr></table>
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(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w0980407.png" /> denotes the trace of a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w0980408.png" />), if the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w0980409.png" /> is positive definite, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804010.png" /> in other cases. The Wishart distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804011.png" /> degrees of freedom and with matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804012.png" /> is defined as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804013.png" />-dimensional distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804014.png" /> with density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804015.png" />. The sample covariance matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804016.png" />, which is an estimator for the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804017.png" />, has a Wishart distribution.
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The joint distribution of the elements from the sample covariance matrix of observations from a multivariate [[Normal distribution|normal distribution]]. Let the results of observations have a  $  p $-
 +
dimensional normal distribution  $  N( \mu , \Sigma ) $
 +
with vector mean  $  \mu $
 +
and covariance matrix $  \Sigma $.  
 +
Then the joint density of the elements of the matrix $  A= \sum _ {i=} 1  ^ {n} ( X _ {i} - \overline{X}\; ) ( X _ {i} - \overline{X}\; )  ^  \prime  $
 +
is given by the formula
  
The Wishart distribution is a basic distribution in multivariate statistical analysis; it is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804018.png" />-dimensional generalization (in the sense above) of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804019.png" />-dimensional [[Chi-squared distribution| "chi-squared" distribution]].
+
$$
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w( n, \Sigma )  =
 +
\frac{| A | ^ {( n- p)/2 }
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e ^ {- \mathop{\rm tr}  ( A \Sigma  ^ {-} 1 )/2 } }{2 ^ {( n- 1) p/2 } \pi ^ {p( p- 1)/4 } | \Sigma | ^ {( n- 1)/2 }
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\prod _ { i= } 1 ^ { p }  \Gamma \left ( n-  
 +
\frac{i}{2}
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  \right ) }
  
If the independent random vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804021.png" /> have Wishart distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804023.png" />, respectively, then the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804024.png" /> has the Wishart distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098040/w09804025.png" />.
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$$
 +
 
 +
( $  \mathop{\rm tr}  M $
 +
denotes the trace of a matrix  $  M $),
 +
if the matrix  $  \Sigma $
 +
is positive definite, and  $  w( n, \Sigma )= 0 $
 +
in other cases. The Wishart distribution with  $  n $
 +
degrees of freedom and with matrix  $  \Sigma $
 +
is defined as the  $  p( n+ 1)/2 $-
 +
dimensional distribution  $  W( n, \Sigma ) $
 +
with density  $  w( n, \Sigma ) $.
 +
The sample covariance matrix  $  S= A/( n- 1) $,
 +
which is an estimator for the matrix  $  \Sigma $,
 +
has a Wishart distribution.
 +
 
 +
The Wishart distribution is a basic distribution in multivariate statistical analysis; it is the  $  p $-
 +
dimensional generalization (in the sense above) of the  $  1 $-
 +
dimensional [[Chi-squared distribution| "chi-squared" distribution]].
 +
 
 +
If the independent random vectors  $  X $
 +
and  $  Y $
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have Wishart distributions $  W( n _ {1} , \Sigma ) $
 +
and $  W ( n _ {2} , \Sigma ) $,  
 +
respectively, then the vector $  X + Y $
 +
has the Wishart distribution $  W( n _ {1} + n _ {2} , \Sigma ) $.
  
 
The Wishart distribution was first used by J. Wishart [[#References|[1]]].
 
The Wishart distribution was first used by J. Wishart [[#References|[1]]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Wishart,  ''Biometrika A'' , '''20'''  (1928)  pp. 32–52</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.W. Anderson,  "An introduction to multivariate statistical analysis" , Wiley  (1958)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Wishart,  ''Biometrika A'' , '''20'''  (1928)  pp. 32–52</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.W. Anderson,  "An introduction to multivariate statistical analysis" , Wiley  (1958)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.M. Khirsagar,  "Multivariate analysis" , M. Dekker  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.J. Muirhead,  "Aspects of multivariate statistical theory" , Wiley  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.M. Khirsagar,  "Multivariate analysis" , M. Dekker  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.J. Muirhead,  "Aspects of multivariate statistical theory" , Wiley  (1982)</TD></TR></table>

Revision as of 08:29, 6 June 2020


The joint distribution of the elements from the sample covariance matrix of observations from a multivariate normal distribution. Let the results of observations have a $ p $- dimensional normal distribution $ N( \mu , \Sigma ) $ with vector mean $ \mu $ and covariance matrix $ \Sigma $. Then the joint density of the elements of the matrix $ A= \sum _ {i=} 1 ^ {n} ( X _ {i} - \overline{X}\; ) ( X _ {i} - \overline{X}\; ) ^ \prime $ is given by the formula

$$ w( n, \Sigma ) = \frac{| A | ^ {( n- p)/2 } e ^ {- \mathop{\rm tr} ( A \Sigma ^ {-} 1 )/2 } }{2 ^ {( n- 1) p/2 } \pi ^ {p( p- 1)/4 } | \Sigma | ^ {( n- 1)/2 } \prod _ { i= } 1 ^ { p } \Gamma \left ( n- \frac{i}{2} \right ) } $$

( $ \mathop{\rm tr} M $ denotes the trace of a matrix $ M $), if the matrix $ \Sigma $ is positive definite, and $ w( n, \Sigma )= 0 $ in other cases. The Wishart distribution with $ n $ degrees of freedom and with matrix $ \Sigma $ is defined as the $ p( n+ 1)/2 $- dimensional distribution $ W( n, \Sigma ) $ with density $ w( n, \Sigma ) $. The sample covariance matrix $ S= A/( n- 1) $, which is an estimator for the matrix $ \Sigma $, has a Wishart distribution.

The Wishart distribution is a basic distribution in multivariate statistical analysis; it is the $ p $- dimensional generalization (in the sense above) of the $ 1 $- dimensional "chi-squared" distribution.

If the independent random vectors $ X $ and $ Y $ have Wishart distributions $ W( n _ {1} , \Sigma ) $ and $ W ( n _ {2} , \Sigma ) $, respectively, then the vector $ X + Y $ has the Wishart distribution $ W( n _ {1} + n _ {2} , \Sigma ) $.

The Wishart distribution was first used by J. Wishart [1].

References

[1] J. Wishart, Biometrika A , 20 (1928) pp. 32–52
[2] T.W. Anderson, "An introduction to multivariate statistical analysis" , Wiley (1958)

Comments

References

[a1] A.M. Khirsagar, "Multivariate analysis" , M. Dekker (1972)
[a2] R.J. Muirhead, "Aspects of multivariate statistical theory" , Wiley (1982)
How to Cite This Entry:
Wishart distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wishart_distribution&oldid=28557
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article