The distribution of Hotelling’sT 2

Pierre Jolicoeur
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Pierre Jolicoeur: University of Montreal, Department of Biological Science

Chapter Chapter 30 in Introduction to Biometry, 1999, pp 266-279 from Springer

Abstract: Abstract 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).

Keywords: Covariance Matrix; Confidence Region; Fork Length; Scatter Diagram; Multivariate Normal Distribution (search for similar items in EconPapers)
Date: 1999
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DOI: 10.1007/978-1-4615-4777-8_31

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