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Hotelling's T-square distribution
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 follows. Suppose
are p×1 column vectors whose entries are real numbers. Let
be their mean. Let the p×p nonnegative-definite matrix
be their "sample variance". (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 T2 may be determined for any matrix of rank at least p.
The reason that this is interesting is that if 
is a random variable with a multivariate normal distribution and 
has a Wishart distribution, and 
and 
are independent, then the probability distribution of T2 is Hotelling's T-square distribution.
The assumptions above are frequently met in practice: it can be shown that if 
, are independent, and 
and are as defined above then has a Wishart distribution with m = n − 1 degrees of freedom and is independent of , and
If, moreover, both distributions are nonsingular, it can be shown that
where F is the F-distribution.
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