Hotelling-T^2-distribution
The continuous probability distribution, concentrated on the positive semi-axis , with density
![]() |
depending on two integer parameters (the number of degrees of freedom) and
,
. For
the Hotelling
-distribution reduces to the Student distribution, and for any
it can be regarded as a multivariate generalization of the Student distribution in the following sense. If a
-dimensional random vector
has the normal distribution with null vector of means and covariance matrix
and if
![]() |
where the random vectors are independent, distributed as
and also independent of
, then the random variable
has the Hotelling
-distribution with
degrees of freedom (
is a column vector and
means transposition). If
, then
![]() |
where the random variable has the Student distribution with
degrees of freedom. If in the definition of the random variable
it is assumed that
has the normal distribution with parameters
and
has the normal distribution with parameters
, then the corresponding distribution is called a non-central Hotelling
-distribution with
degrees of freedom and non-centrality parameter
.
Hotelling's -distribution is used in mathematical statistics in the same situation as Student's
-distribution, but then in the multivariate case (see Multi-dimensional statistical analysis). If the results of observations
are independent normally-distributed random vectors with mean vector
and non-degenerate covariance matrix
, then the statistic
![]() |
where
![]() |
and
![]() |
has the Hotelling -distribution with
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
-distribution, because the random variable
has the
-distribution with
and
degrees of freedom.
The Hotelling -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) |
Hotelling-T^2-distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hotelling-T%5E2-distribution&oldid=12614