The distribution of Hotelling’sT 2

When several variates X1,..., Xi, Xj,...,Xq follow a multivariate normal distribution, the quadratic form $$(\operatorname{X} - \mu ){\sum ^{ - 1}}\left( {\tilde X - \tilde \mu } \right) $$ , which appears in the q-dimensional normal probability density, follows a χ 2 distribution with q degrees of freedom (section 29.1). In theory, the χ 2 distribution could thus be used to test hypotheses or to delimit confidence regions concerning the mean vector it if the parametric covariance matrix Σ were known. In practice, however, the population covariance matrix Σ is seldom known and is generally replaced by its estimate, the sample covariance matrix S (section 29.3).