The distribution of Hotelling’sT ²

Abstract

When several variates X₁,..., X_i, X_j,...,X_q 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).