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'' $  T  ^ {2} $-test''

A test intended for testing a hypothesis  $  H _ {0} $
according to which the true value of the unknown vector  $  \mu = ( \mu _ {1}, \dots, \mu _ {p} ) $
of mathematical expectation of a non-degenerate  $  p $-dimensional normal law  $  N ( \mu , B) $
whose covariance matrix  $  B $
is also unknown, is the vector  $  \mu = ( \mu _ {10}, \dots, \mu _ {p0} ) $. 
Hotelling's test is based on the following result. Let  $  X _ {1}, \dots, X _ {n} $
be independent  $  p $-dimensional random vectors,  $  n - 1 \geq  p $, 
subject to the non-degenerate normal law  $  N ( \mu , B) $, 
and let

$$ 
T  ^ {2}  = \ 
n ( \overline{X} - \mu _ {0} ) ^ {T }
S  ^ {-1} ( \overline{X} - \mu _ {0} ),
$$

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} ) ^ {T }
$$

are maximum-likelihood estimators for the unknown parameters  $  \mu $
and  $  B $. 
Then the statistic

$$ 
F  = \ 

\frac{n - p }{p ( n - 1) }

T  ^ {2}
$$

has the non-central [[Fisher-F-distribution|Fisher  $  F $-distribution]] with  $  p $
and  $  n - p $
degrees of freedom and non-centrality parameter

$$ 
n ( \mu - \mu _ {0} ) ^ {T } B  ^ {-1} ( \mu - \mu _ {0} );
$$

the statistic  $  T  ^ {2} $
has the [[Hotelling-T^2-distribution|Hotelling  $  T  ^ {2} $-distribution]]. Consequently, to test the hypothesis  $  H _ {0} $: 
$  \mu = \mu _ {0} $
against the alternative  $  H _ {1} $: 
$  \mu \neq \mu _ {0} $
one can compute the values of the statistic  $  F $
based on realizations of the independent random vectors  $  X _ {1}, \dots, X _ {n} $
from the non-degenerate  $  p $-dimensional normal law  $  N ( \mu , B) $, 
which under the hypothesis  $  H _ {0} $
has the central  $  F $-distribution with  $  p $
and  $  n - p $
degrees of freedom. Using Hotelling's test with significance level  $  \alpha $, 
$  H _ {0} $
must be rejected if  $  F \geq  F _  \alpha  ( p, n - p) $, 
where  $  F _  \alpha  ( p, n - p) $
is the  $  \alpha $-quantile of the  $  F $-distribution. The connection between Hotelling's test and the generalized likelihood-ratio test should be mentioned. Let

$$ 
L ( \mu , B)  = \ 
L ( X _ {1}, \dots, X _ {n} ;  \mu , B) =
$$

$$ 
= \ 

\frac{| B  ^ {-1} |  ^ {n/2} }{( 2 \pi )  ^ {np/2}
 }
  \mathop{\rm exp} \left \{ - {
\frac{1}{2}
 } \sum _ {i = 1 } ^ { n }  ( X _ {i} - \mu ) ^ {T } B  ^ {-1} ( X _ {i} - \mu ) \right \}
$$

be the likelihood function computed from the sample  $  X _ {1}, \dots, X _ {n} $. 
The generalized likelihood-ratio test for testing the simple hypothesis  $  H _ {0} $: 
$  \mu = \mu _ {0} $
against the compound alternative  $  H _ {1} $: 
$  \mu \neq \mu _ {0} $
is constructed from the statistic

$$ 
\lambda  = \ 
\lambda ( X _ {1}, \dots, X _ {n} )  = \ 

\frac{\sup _ { B }  L ( \mu _ {0} , B) }{\sup _ {\mu , B }  L ( \mu , B) }
 .
$$

The statistic  $  \lambda $
and the statistics  $  T  ^ {2} $
and  $  F $
are related by:

$$ 
\lambda  ^ {2/n}  = \ 

\frac{n - 1 }{T  ^ {2} + n - 1 }
  = \ 

\frac{n - p }{pF + n - p }
 .
$$

For testing the hypothesis  $  H _ {0} $: 
$  \mu = \mu _ {0} $, 
Hotelling's test is uniformly most powerful among all tests that are invariant under similarity transformations (see [[Most-powerful test|Most-powerful test]]; [[Invariant test|Invariant test]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T.W. Anderson,   "An introduction to multivariate statistical analysis" , Wiley  (1984)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.R. Rao,   "Linear statistical inference and its applications" , Wiley  (1973)</TD></TR></table>