A Tutorial on Multivariate Statistical Analysis

A Tutorial onMultivariateStatistical AnalysisCraig A. TracyUC DavisSAMSIS eptember 20061 ELEMENTARY STATISTICSC ollection of (real-valued) data from a sequence of experimentsX1,X2,..,XnMight make assumption underlying law isN( , 2) with unknownmean and variance 2. Want to estimate and 2from the Mean & Sample Variance: X=1nXjXj, S=1n 1Xj Xj X 2 Estimators are unbiased E( X) = ,E(S) = 22 Theorem:IfX1,X2,..are independentN( , 2) variables then XandSare independent. We have that XisN( , 2/n) and(n 1)S/ 2is 2(n 1).Recall 2(d) denotes the chi-squared distribution withddegrees offreedom. Its density isf 2(x) =12d/2 (d/2)xd/2 1e x/2, x 0,where (z) =Z 0tz 1e tdt, (z)> GENERALIZATIONSFrom the classic textbook of Anderson[1]: Multivariate Statistical Analysis is concerned with data thatconsists of sets of measurements on a number of individualsor objects.

•The Wishart distribution is the multivariate generalization of the chi-squared distribution. •A∼W p (n,Σ) is positive definite with probability one if and

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