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.
If the vector pd1 is Gaussian multivariate-distributed with zero mean and unit covariance matrix N(p01,pIp) and mMp is a p x p matrix with a Wishart distribution with unit scale matrix and m degrees of freedom W(pIp,m) then m(1d' pM−1pd1) has a Hotelling T2 distribution with dimensionality parameter p and m degrees of freedom.
If the notation is used to denote a random variable having Hotelling's T-squared distribution with parameters p and m then, if a random variable X has Hotelling's T-squared distribution,
where is the F-distribution with parameters p and m−p+1.
Hotelling's T-squared statistic
be n independent random variables, which may be represented as column vectors of real numbers. Define
to be the sample mean. It can be shown that
where is the chi-squared distribution with p degrees of freedom. To show this use the fact that and then derive the characteristic function of the random variable . This is done below,
However, is often unknown and we wish to do hypothesis testing on the location .
Sum of p squared t's
to be the sample covariance. Here we denote transpose by an apostrophe. It can be shown that is positive-definite and follows a p-variate Wishart distribution with n−1 degrees of freedom. Hotelling's T-squared statistic is then defined to be
and, also from above,
where is the F-distribution with parameters p and n−p. In order to calculate a p value, multiply the t2 statistic by the above constant and use the F-distribution.
Hotelling's two-sample T-squared statistic
as the sample means, and
as the unbiased pooled covariance matrix estimate, then Hotelling's two-sample T-squared statistic is
and it can be related to the F-distribution by
where is the difference vector between the population means.
- Student's t-test in univariate statistics
- Student's t-distribution in univariate probability theory
- Multivariate Student distribution.
- F-distribution (commonly tabulated or available in software libraries, and hence used for testing the T-squared statistic using the relationship given above)
- Wilks' lambda distribution (in multivariate statistics Wilks's Λ is to Hotelling's T2 as Snedecor's F is to Student's t in univariate statistics).
- Hotelling, H. (1931). "The generalization of Student's ratio". Annals of Mathematical Statistics 2 (3): 360–378. doi:10.1214/aoms/1177732979.
- Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, Second Edition, Chapman & Hall/CRC, 2003, p. 1408
- K.V. Mardia, J.T. Kent, and J.M. Bibby (1979) Multivariate Analysis, Academic Press.
- Prokhorov, A.V. (2001), "Hotelling T2-distribution", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4