Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions. I. The Equal-Means Case

We consider the problem of discriminating, on the basis of random “training” samples, between two independent multivariate normal populations, Np(μ, Σ1) and Np(μ, Σ2), which have a common mean vector μ and distinct covariance matrices Σ1 and Σ2. Using the theory of Bessel functions of the second kind of matrix argument developed by Herz (1955, Ann. Math.61, 474–523), 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, Σ1 and Σ2 we obtain explicit formulas and inequalities for the probabilities of misclassification. We apply these results to data given by Stocks (1933, Ann. Eugen.5, 1–55) in a biometric investigation of the physical characteristics of twins, and to data provided by Rencher (1995, “Methods of Multivariate Analysis,” Wiley, New York) 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 Stocksʹ data we make extensive comparisons with previously published estimates.