Hotelling's T-squared distribution

In statistics Hotelling's T-squared distribution is a univariate distribution proportional to the F-distribution and arises importantly as the distribution of a set of statistics which are natural generalizations of the statistics underlying Student's t-distribution. In particular, the distribution arises in multivariate statistics in undertaking tests of the differences between the (multivariate) means of different populations, where tests for univariate problems would make use of a t-test. The distribution is named for Harold Hotelling, who developed it as a generalization of Student's t-distribution.

If the vector _pd₁ is Gaussian multivariate-distributed with zero mean and unit covariance matrix N(_p0₁,_pI_p) and _pM_p is a p x p matrix with unit scale matrix and m degrees of freedom with a Wishart distribution W(_pI_p,m), then the Quadratic form m(₁d^T _pM⁻¹_pd₁) has a Hotelling T²(p,m) distribution with dimensionality parameter p and m degrees of freedom.

If a random variable X has Hotelling's T-squared distribution, ${\displaystyle X\sim T_{p,m}^{2}}$, then:

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