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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I: The equal-means case. (English) Zbl 1098.62530
Summary: We consider the problem of discriminating, on the basis of random ‘training’ samples, between two independent multivariate normal populations, N p (μ,Σ 1 ) and N p (μ,Σ 2 ), which have a common mean vector μ and distinct covariance matrices Σ 1 and Σ 2 . Using the theory of Bessel functions of the second kind of matrix argument developed by C. S. Herz [Ann. Math. 61, 474–523 (1955; Zbl 0066.32002)], we derive stochastic representations for the exact distributions of the ‘plug-in’ quadratic discriminant functions for classifying a newly obtained observation. These stochastic representations involve only chi-squared and F-distributions, hence we obtain an efficient method for simulating the discriminant functions and estimating the corresponding probabilities of misclassification. For some special values of p, Σ 1 and Σ 2 we obtain explicit formulas and inequalities for the probabilities of misclassification. We apply these results to data given by Stocks [Ann. Eugen. 5, 1–55 (1933)] in a biometric investigation of the physical characteristics of twins, and to data provided by A. C. Rencher [Methods of Multivariate Analysis. (1995; Zbl 0836.62039)] in a study of the relationship between football helmet design and neck injuries. For each application we estimate the exact probabilities of misclassification, and in the case of Stocks’ data we make extensive comparisons with previously published estimates.

MSC:
62H10Multivariate distributions of statistics
62H30Classification and discrimination; cluster analysis (statistics)
62E15Exact distribution theory in statistics