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Difference between revisions of "Hotelling test"

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m (tex encoded by computer)
m (fixing superscripts)
 
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{{TEX|done}}
 
{{TEX|done}}
  
'' $  T  ^ {2} $-
+
'' $  T  ^ {2} $-test''
test''
 
  
 
A test intended for testing a hypothesis  $  H _ {0} $
 
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} ) $
+
according to which the true value of the unknown vector  $  \mu = ( \mu _ {1}, \dots, \mu _ {p} ) $
of mathematical expectation of a non-degenerate  $  p $-
+
of mathematical expectation of a non-degenerate  $  p $-dimensional normal law  $  N ( \mu , B) $
dimensional normal law  $  N ( \mu , B) $
 
 
whose covariance matrix  $  B $
 
whose covariance matrix  $  B $
is also unknown, is the vector  $  \mu = ( \mu _ {10} \dots \mu _ {p0} ) $.  
+
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} $
+
Hotelling's test is based on the following result. Let  $  X _ {1}, \dots, X _ {n} $
be independent  $  p $-
+
be independent  $  p $-dimensional random vectors,  $  n - 1 \geq  p $,  
dimensional random vectors,  $  n - 1 \geq  p $,  
 
 
subject to the non-degenerate normal law  $  N ( \mu , B) $,  
 
subject to the non-degenerate normal law  $  N ( \mu , B) $,  
 
and let
 
and let
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$$  
 
$$  
 
T  ^ {2}  = \  
 
T  ^ {2}  = \  
n ( \overline{X}\; - \mu _ {0} ) ^ {T }
+
n ( \overline{X} - \mu _ {0} ) ^ {T }
S  ^ {-} 1 ( \overline{X}\; - \mu _ {0} ),
+
S  ^ {-1} ( \overline{X} - \mu _ {0} ),
 
$$
 
$$
  
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\sum _ {i = 1 } ^ { n }  
 
\sum _ {i = 1 } ^ { n }  
( X _ {i} - \overline{X}\; ) ( X _ {i} - \overline{X}\; ) ^ {T }
+
( X _ {i} - \overline{X} ) ( X _ {i} - \overline{X} ) ^ {T }
 
$$
 
$$
  
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$$
 
$$
  
has the non-central [[Fisher-F-distribution|Fisher  $  F $-
+
has the non-central [[Fisher-F-distribution|Fisher  $  F $-distribution]] with  $  p $
distribution]] with  $  p $
 
 
and  $  n - p $
 
and  $  n - p $
 
degrees of freedom and non-centrality parameter
 
degrees of freedom and non-centrality parameter
  
 
$$  
 
$$  
n ( \mu - \mu _ {0} ) ^ {T } B  ^ {-} 1 ( \mu - \mu _ {0} );
+
n ( \mu - \mu _ {0} ) ^ {T } B  ^ {-1} ( \mu - \mu _ {0} );
 
$$
 
$$
  
 
the statistic  $  T  ^ {2} $
 
the statistic  $  T  ^ {2} $
has the [[Hotelling-T^2-distribution|Hotelling  $  T  ^ {2} $-
+
has the [[Hotelling-T^2-distribution|Hotelling  $  T  ^ {2} $-distribution]]. Consequently, to test the hypothesis  $  H _ {0} $:  
distribution]]. Consequently, to test the hypothesis  $  H _ {0} $:  
 
 
$  \mu = \mu _ {0} $
 
$  \mu = \mu _ {0} $
 
against the alternative  $  H _ {1} $:  
 
against the alternative  $  H _ {1} $:  
 
$  \mu \neq \mu _ {0} $
 
$  \mu \neq \mu _ {0} $
 
one can compute the values of the statistic  $  F $
 
one can compute the values of the statistic  $  F $
based on realizations of the independent random vectors  $  X _ {1} \dots X _ {n} $
+
based on realizations of the independent random vectors  $  X _ {1}, \dots, X _ {n} $
from the non-degenerate  $  p $-
+
from the non-degenerate  $  p $-dimensional normal law  $  N ( \mu , B) $,  
dimensional normal law  $  N ( \mu , B) $,  
 
 
which under the hypothesis  $  H _ {0} $
 
which under the hypothesis  $  H _ {0} $
has the central  $  F $-
+
has the central  $  F $-distribution with  $  p $
distribution with  $  p $
 
 
and  $  n - p $
 
and  $  n - p $
 
degrees of freedom. Using Hotelling's test with significance level  $  \alpha $,  
 
degrees of freedom. Using Hotelling's test with significance level  $  \alpha $,  
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must be rejected if  $  F \geq  F _  \alpha  ( p, n - p) $,  
 
must be rejected if  $  F \geq  F _  \alpha  ( p, n - p) $,  
 
where  $  F _  \alpha  ( p, n - p) $
 
where  $  F _  \alpha  ( p, n - p) $
is the  $  \alpha $-
+
is the  $  \alpha $-quantile of the  $  F $-distribution. The connection between Hotelling's test and the generalized likelihood-ratio test should be mentioned. Let
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 ( \mu , B)  = \  
L ( X _ {1} \dots X _ {n} ;  \mu , B) =
+
L ( X _ {1}, \dots, X _ {n} ;  \mu , B) =
 
$$
 
$$
  
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= \  
 
= \  
  
\frac{| B  ^ {-} 1 |  ^ {n/2} }{( 2 \pi )  ^ {np/2}
+
\frac{| B  ^ {-1} |  ^ {n/2} }{( 2 \pi )  ^ {np/2}
 
  }
 
  }
 
   \mathop{\rm exp} \left \{ - {
 
   \mathop{\rm exp} \left \{ - {
 
\frac{1}{2}
 
\frac{1}{2}
  } \sum _ {i = 1 } ^ { n }  ( X _ {i} - \mu ) ^ {T } B  ^ {-} 1 ( X _ {i} - \mu ) \right \}
+
  } \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} $.  
+
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} $:  
 
The generalized likelihood-ratio test for testing the simple hypothesis  $  H _ {0} $:  
 
$  \mu = \mu _ {0} $
 
$  \mu = \mu _ {0} $
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$$  
 
$$  
 
\lambda  = \  
 
\lambda  = \  
\lambda ( X _ {1} \dots X _ {n} )  = \  
+
\lambda ( X _ {1}, \dots, X _ {n} )  = \  
  
 
\frac{\sup _ { B }  L ( \mu _ {0} , B) }{\sup _ {\mu , B }  L ( \mu , B) }
 
\frac{\sup _ { B }  L ( \mu _ {0} , B) }{\sup _ {\mu , B }  L ( \mu , B) }

Latest revision as of 01:40, 5 March 2022


$ 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 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. 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; Invariant test).

References

[1] T.W. Anderson, "An introduction to multivariate statistical analysis" , Wiley (1984)
[2] C.R. Rao, "Linear statistical inference and its applications" , Wiley (1973)
How to Cite This Entry:
Hotelling test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hotelling_test&oldid=47275
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article