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Zbl 1098.62530
McFarland, H. Richard III; Richards, Donald St. P.
Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I: The equal-means case. (English)
[J] J. Multivariate Anal. 77, No. 1, 21-53 (2001). ISSN 0047-259X

Summary: We consider the problem of discriminating, on the basis of random `training' samples, between two independent multivariate normal populations, $N_p(\mu, \Sigma_1)$ and $N_p(\mu, \Sigma_2)$, which have a common mean vector $\mu$ and distinct covariance matrices $\Sigma_1$ and $\Sigma_2$. Using the theory of Bessel functions of the second kind of matrix argument developed by {\it 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, $\Sigma_1$ and $\Sigma_2$ 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 {\it 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 2000:: *62H10 Multivariate distributions of statistics
62H30 Statistical classification, etc.
62E15 Exact distribution theory in statistics

Keywords: Bessel function of matrix argument; biplot; discriminant analysis; dizygotic twins; football helmet design; misclassification probability; monozygotic twins; multivariate gamma function; multivariate normality; neck injuries; stochastic representation; stochastic ordering; Wishart distribution

Citations: Zbl 0066.32002; Zbl 0836.62039

Cited in: Zbl 1099.62517

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