# 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)
How to Cite This Entry:
Hotelling-T^2-distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hotelling-T%5E2-distribution&oldid=47274
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article