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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II: The heterogeneous case. (English) Zbl 1099.62517
Summary: We consider the problem of discriminating between two independent multivariate normal populations, N p (μ,Σ 1 ) and N p (μ,Σ 2 ) having distinct mean vectors μ 1 and μ 2 and distinct covariance matrices Σ 1 and Σ 2 . The parameters μ 1 , μ 2 , Σ 1 , Σ 2 are unknown and are estimated by means of independent random training samples from each population. We derive a stochastic representation for the exact distribution of the ‘plug-in’ quadratic discriminant function for classifying a new observation between the two populations. The stochastic representation involves only the classical standard normal, chi-square, and F distributions and is easily implemented for simulation purposes. Using Monte Carlo simulation of the stochastic representation we provide applications to the estimation of misclassification probabilities for the well-known iris data studied by Fisher [Ann. Eugen. 7, 179–188 (1936)]; a data set on corporate financial ratios provided by R. A. Johnson and D. W. Wichern [Applied Multivariate Statistical Analysis, 4th ed., Prentice-Hall, Englewood Cliffs, NJ (1998), see Zbl 0745.62050]; and a data set analyzed by Reaven and Miller [Diabetologia 16, 17–24 (1979)] in a classification of diabetic status. For part I see J. Multivariate Anal. 77, No. 1, 21–53 (2001; Zbl 1098.62517).
MSC:
62H10Multivariate distributions of statistics
62H30Classification and discrimination; cluster analysis (statistics)
62E15Exact distribution theory in statistics