In statistics, Hotelling's T-square statistic, named for Harold Hotelling,
is a generalization of Student's t statistic that is used in multivariate hypothesis testing.
Hotelling's T-square statistic is defined as
where n is a number of points (see below), is a column vector of elements and is a matrix.
If is a random variable with a multivariate Gaussian distribution and (independent of x) has a Wishart distribution with the same non-singular variance matrix and with ,
then the distribution
of is , Hotelling's T-square distribution with parameters p and m.
It can be shown that
where is the F-distribution.
Now suppose that
are p×1 column vectors whose entries are real numbers. Let
be their mean. Let the p×p positive-definite matrix
be their "sample variance" matrix. (The transpose of any matrix M is denoted above by M′). Let μ be some known p×1 column vector (in applications a hypothesized value of a population mean). Then Hotelling's T-square statistic is
Note that is closely related to the squared Mahalanobis distance.
In particular, it can be shown
that if , are independent, and and are as defined above then has a Wishart distribution with n − 1 degrees of freedom
and is independent of , and
This implies that:
Hotelling's two-sample T-square statistic
If and , with the samples independently drawn from two independent multivariate normal distributions with the same mean and covariance, and we define
as the sample means, and
as the unbiased pooled covariance matrix estimate, then Hotelling's two-sample T-square statistic is
and it can be related to the F-distribution by
- ↑ H. Hotelling (1931) The generalization of Student's ratio, Ann. Math. Statist., Vol. 2, pp360-378.
- ↑ 2.0 2.1 K.V. Mardia, J.T. Kent, and J.M. Bibby (1979) Multivariate Analysis, Academic Press.
it:Variabile casuale T-quadrato di Hotelling