View source for Hotelling-T^2-distribution

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

====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>