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, Np([mu]1, [Sigma]1) and Np([mu]2, [Sigma]2), having distinct mean vectors [mu]1 and [mu]2 and distinct covariance matrices [Sigma]1 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 (Diabetologia16 (1979), 17-24) in a classification of diabetic status.