Hotelling test
-
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) |
Hotelling test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hotelling_test&oldid=47275