Hotelling's T-squared Distribution - Hotelling's T-squared Statistic

Hotelling's T-squared Statistic

Hotelling's T-squared statistic is a generalization of Student's t statistic that is used in multivariate hypothesis testing, and is defined as follows.

Let denote a -variate normal distribution with location and covariance . Let

be 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 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 .

Define

to be the sample covariance. Here we denote transpose by an apostrophe. It can be shown that is positive-definite and follows a -variate Wishart distribution with degrees of freedom. Hotelling's T-squared statistic is then defined to be

because it can be shown that

i.e.

where is the F-distribution with parameters and . In order to calculate a p value, multiply the statistic by the above constant and use the F distribution.