A Tutorial on Multivariate Statistical Analysis

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A Tutorial onMultivariateStatistical AnalysisCraig A. TracyUC DavisSAMSISeptember 20061ELEMENTARY STATISTICSCollection 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 2Estimators are unbiased E( X) = ,E(S) = 22Theorem: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∼Wp(n,Σ) is positive definite with probability one if and only if n≥p. •The sample covariance matrix, S= 1 n−1 A is Wp(n−1, 1 n−1 Σ). 10

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