# Hotelling test

$T ^ {2}$- 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).

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