Hotelling's T-squared distribution

In statistics, particularly in hypothesis testing, the Hotelling's T-squared distribution (T²), proposed by Harold Hotelling, is a multivariate probability distribution that is tightly related to the F-distribution and is most notable for arising as the distribution of a set of sample statistics that are natural generalizations of the statistics underlying the Student's t-distribution. The Hotelling's t-squared statistic (t²) is a generalization of Student's t-statistic that is used in multivariate hypothesis testing.

Hotelling's T² distribution

Probability density function
Cumulative distribution function
Parameters	p - dimension of the random variables m - related to the sample size
Support	${\displaystyle x\in (0,+\infty )\;}$ if ${\displaystyle p=1}$ ${\displaystyle x\in [0,+\infty )\;}$ otherwise.