Exact misclassification probabilities for plug-in normal quadratic discriminant functions II. The heterogeneous case

We consider the problem of discriminating between two independent multivariate normal populations, N-p(mu(1), Sigma(1)) and N-p(mu(2), Sigma(2)) having distinct mean vectors mu(1) and mu(2) and distinct covariance matrices Sigma(2), and Sigma(2). The parameters mu(1), mu(2), Sigma(1), and Sigma(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 (1936), 179-188); a data set on corporate financial ratios provided by Johnson and Wichern (Applied Multivariate Statistical Analysis, 4th ed., Prentice-Hall, Englewood Cliffs, NJ, 1998); and a data set analyzed by Reaven and Miller (Diabetologia 16 (1979), 17-24) in a classification of diabetic status. (C) 2002 Elsevier Science (USA).