Hotelling-T^2-distribution

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, 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). 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 or of the 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 [1] for testing equality of means of two normal populations.

References

[1]	H. Hotelling, "The generalization of Student's ratio" Ann. Math. Stat. , 2 (1931) pp. 360–378
[2]	T.W. Anderson, "An introduction to multivariate statistical analysis" , Wiley (1984)