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The continuous probability distribution, concentrated on the positive semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h0480702.png" />, with density
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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/h/h048/h048070/h0480703.png" /></td> </tr></table>
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depending on two integer parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h0480704.png" /> (the number of degrees of freedom) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h0480705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h0480706.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h0480707.png" /> the Hotelling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h0480708.png" />-distribution reduces to the [[Student distribution|Student distribution]], and for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h0480709.png" /> it can be regarded as a multivariate generalization of the Student distribution in the following sense. If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807010.png" />-dimensional random vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807011.png" /> has the normal distribution with null vector of means and covariance matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807012.png" /> and if
+
The continuous probability distribution, concentrated on the positive semi-axis  $  ( 0, \infty ) $,  
 +
with density
  
<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/h/h048/h048070/h04807013.png" /></td> </tr></table>
+
$$
 +
p ( x)  = \
  
where the random vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807014.png" /> are independent, distributed as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807015.png" /> and also independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807016.png" />, then the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807017.png" /> has the Hotelling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807018.png" />-distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807019.png" /> degrees of freedom (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807020.png" /> is a column vector and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807021.png" /> means transposition). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807022.png" />, then
+
\frac{\Gamma ( {( n + 1) / 2 } )
 +
x ^ { {k / 2 } - 1 }
 +
( 1 + {x / n } ) ^ {- {( n + 1) / 2 } } }{\Gamma ( {( n - k + 1) / 2 } )
 +
\Gamma ( {k / 2 } ) n ^ { {k / 2 } } }
 +
,
 +
$$
  
<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/h/h048/h048070/h04807023.png" /></td> </tr></table>
+
depending on two integer parameters  $  n $(
 +
the number of degrees of freedom) and  $  k $,
 +
$  n \geq  k \geq  1 $.
 +
For  $  k = 1 $
 +
the Hotelling  $  T  ^ {2} $-
 +
distribution reduces to the [[Student distribution|Student distribution]], and for any  $  k > 0 $
 +
it can be regarded as a multivariate generalization of the Student distribution in the following sense. If a  $  k $-
 +
dimensional random vector  $  Y $
 +
has the normal distribution with null vector of means and covariance matrix  $  \Sigma $
 +
and if
  
where the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807024.png" /> has the Student distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807025.png" /> degrees of freedom. If in the definition of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807026.png" /> it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807027.png" /> has the normal distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807029.png" /> has the normal distribution with parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807030.png" />, then the corresponding distribution is called a non-central Hotelling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807032.png" />-distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807033.png" /> degrees of freedom and non-centrality parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807034.png" />.
+
$$
 +
= {
 +
\frac{1}{n}
 +
}
 +
\sum _ {i = 1 } ^ { n }
 +
Z _ {i} ^ { \prime }
 +
Z _ {i} ,
 +
$$
  
Hotelling's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807035.png" />-distribution is used in mathematical statistics in the same situation as Student's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807036.png" />-distribution, but then in the multivariate case (see [[Multi-dimensional statistical analysis|Multi-dimensional statistical analysis]]). If the results of observations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807037.png" /> are independent normally-distributed random vectors with mean vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807038.png" /> and non-degenerate covariance matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807039.png" />, then the statistic
+
where the random vectors  $  Z _ {i} $
 +
are independent, distributed as $  Y $
 +
and also independent of  $  Y $,  
 +
then the random variable  $  T  ^ {2} = Y ^ { \prime } S  ^ {-} 1 Y $
 +
has the Hotelling  $  T  ^ {2} $-
 +
distribution with $  n $
 +
degrees of freedom ( $  Y $
 +
is a column vector and  $  {}  ^  \prime  $
 +
means transposition). If  $  k = 1 $,  
 +
then
  
<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/h/h048/h048070/h04807040.png" /></td> </tr></table>
+
$$
 +
T  ^ {2}  = \
 +
 
 +
\frac{Y  ^ {2} }{\chi _ {n}  ^ {2} /n }
 +
  = \
 +
t _ {n}  ^ {2} ,
 +
$$
 +
 
 +
where the random variable  $  t _ {n} $
 +
has the Student distribution with  $  n $
 +
degrees of freedom. If in the definition of the random variable  $  T  ^ {2} $
 +
it is assumed that  $  Y $
 +
has the normal distribution with parameters  $  ( \nu , \Sigma ) $
 +
and  $  Z _ {i} $
 +
has the normal distribution with parameters  $  ( 0, \Sigma ) $,
 +
then the corresponding distribution is called a non-central Hotelling  $  T  ^ {2} $-
 +
distribution with  $  n $
 +
degrees of freedom and non-centrality parameter  $  \nu $.
 +
 
 +
Hotelling's  $  T  ^ {2} $-
 +
distribution is used in mathematical statistics in the same situation as Student's  $  t $-
 +
distribution, but then in the multivariate case (see [[Multi-dimensional statistical analysis|Multi-dimensional statistical analysis]]). If the results of observations  $  X _ {1} \dots X _ {n} $
 +
are independent normally-distributed random vectors with mean vector  $  \nu $
 +
and non-degenerate covariance matrix  $  \Sigma $,
 +
then the statistic
 +
 
 +
$$
 +
T  ^ {2}  = \
 +
n ( \overline{X}\; - \mu )  ^  \prime
 +
S  ^ {-} 1 ( \overline{X}\; - \mu ),
 +
$$
  
 
where
 
where
  
<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/h/h048/h048070/h04807041.png" /></td> </tr></table>
+
$$
 +
\overline{X}\; = {
 +
\frac{1}{n}
 +
}
 +
\sum _ {i = 1 } ^ { n }
 +
X _ {i}  $$
  
 
and
 
and
  
<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/h/h048/h048070/h04807042.png" /></td> </tr></table>
+
$$
 +
= {
 +
\frac{1}{n - 1 }
 +
}
 +
\sum _ {i = 1 } ^ { n }
 +
( X _ {i} - \overline{X}\; )
 +
( X _ {i} - \overline{X}\; )  ^  \prime  ,
 +
$$
  
has the Hotelling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807043.png" />-distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807044.png" /> degrees of freedom. This fact forms the basis of the [[Hotelling test|Hotelling test]]. For numerical calculations one uses tables of the [[Beta-distribution|beta-distribution]] or of the [[Fisher-F-distribution|Fisher <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807045.png" />-distribution]], because the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807046.png" /> has the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807047.png" />-distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807049.png" /> degrees of freedom.
+
has the Hotelling $  T  ^ {2} $-
 +
distribution with $  n - 1 $
 +
degrees of freedom. This fact forms the basis of the [[Hotelling test|Hotelling test]]. For numerical calculations one uses tables of the [[Beta-distribution|beta-distribution]] or of the [[Fisher-F-distribution|Fisher $  F $-
 +
distribution]], because the random variable $  (( n - k + 1)/nk) T  ^ {2} $
 +
has the $  F $-
 +
distribution with $  k $
 +
and $  n - k + 1 $
 +
degrees of freedom.
  
The Hotelling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048070/h04807050.png" />-distribution was proposed by H. Hotelling [[#References|[1]]] for testing equality of means of two normal populations.
+
The Hotelling $  T  ^ {2} $-
 +
distribution was proposed by H. Hotelling [[#References|[1]]] for testing equality of means of two normal populations.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hotelling,  "The generalization of Student's ratio"  ''Ann. Math. Stat.'' , '''2'''  (1931)  pp. 360–378</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.W. Anderson,  "An introduction to multivariate statistical analysis" , Wiley  (1984)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hotelling,  "The generalization of Student's ratio"  ''Ann. Math. Stat.'' , '''2'''  (1931)  pp. 360–378</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T.W. Anderson,  "An introduction to multivariate statistical analysis" , Wiley  (1984)</TD></TR></table>

Revision as of 22:11, 5 June 2020


The continuous probability distribution, concentrated on the positive semi-axis $ ( 0, \infty ) $, with density

$$ p ( x) = \ \frac{\Gamma ( {( n + 1) / 2 } ) x ^ { {k / 2 } - 1 } ( 1 + {x / n } ) ^ {- {( n + 1) / 2 } } }{\Gamma ( {( n - k + 1) / 2 } ) \Gamma ( {k / 2 } ) n ^ { {k / 2 } } } , $$

depending on two integer parameters $ n $( the number of degrees of freedom) and $ k $, $ n \geq k \geq 1 $. For $ k = 1 $ the Hotelling $ T ^ {2} $- distribution reduces to the Student distribution, and for any $ k > 0 $ it can be regarded as a multivariate generalization of the Student distribution in the following sense. If a $ k $- dimensional random vector $ Y $ has the normal distribution with null vector of means and covariance matrix $ \Sigma $ and if

$$ S = { \frac{1}{n} } \sum _ {i = 1 } ^ { n } Z _ {i} ^ { \prime } Z _ {i} , $$

where the random vectors $ Z _ {i} $ are independent, distributed as $ Y $ and also independent of $ Y $, then the random variable $ T ^ {2} = Y ^ { \prime } S ^ {-} 1 Y $ has the Hotelling $ T ^ {2} $- distribution with $ n $ degrees of freedom ( $ Y $ is a column vector and $ {} ^ \prime $ means transposition). If $ k = 1 $, then

$$ T ^ {2} = \ \frac{Y ^ {2} }{\chi _ {n} ^ {2} /n } = \ t _ {n} ^ {2} , $$

where the random variable $ t _ {n} $ has the Student distribution with $ n $ degrees of freedom. If in the definition of the random variable $ T ^ {2} $ it is assumed that $ Y $ has the normal distribution with parameters $ ( \nu , \Sigma ) $ and $ Z _ {i} $ has the normal distribution with parameters $ ( 0, \Sigma ) $, then the corresponding distribution is called a non-central Hotelling $ T ^ {2} $- distribution with $ n $ degrees of freedom and non-centrality parameter $ \nu $.

Hotelling's $ T ^ {2} $- distribution is used in mathematical statistics in the same situation as Student's $ t $- distribution, but then in the multivariate case (see Multi-dimensional statistical analysis). If the results of observations $ X _ {1} \dots X _ {n} $ are independent normally-distributed random vectors with mean vector $ \nu $ and non-degenerate covariance matrix $ \Sigma $, then the statistic

$$ T ^ {2} = \ n ( \overline{X}\; - \mu ) ^ \prime S ^ {-} 1 ( \overline{X}\; - \mu ), $$

where

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

and

$$ S = { \frac{1}{n - 1 } } \sum _ {i = 1 } ^ { n } ( X _ {i} - \overline{X}\; ) ( X _ {i} - \overline{X}\; ) ^ \prime , $$

has the Hotelling $ T ^ {2} $- distribution with $ n - 1 $ degrees of freedom. This fact forms the basis of the Hotelling test. For numerical calculations one uses tables of the beta-distribution or of the Fisher $ F $- distribution, because the random variable $ (( n - k + 1)/nk) T ^ {2} $ has the $ F $- distribution with $ k $ and $ n - k + 1 $ degrees of freedom.

The Hotelling $ T ^ {2} $- distribution was proposed by H. Hotelling [1] for testing equality of means of two normal populations.

References

[1] H. Hotelling, "The generalization of Student's ratio" Ann. Math. Stat. , 2 (1931) pp. 360–378
[2] T.W. Anderson, "An introduction to multivariate statistical analysis" , Wiley (1984)
How to Cite This Entry:
Hotelling-T^2-distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hotelling-T%5E2-distribution&oldid=12614
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article