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(μ₁, ∑₁) and N_p(μ₂, ∑₂), having distinct mean vectors μ₁ and μ₂ and distinct covariance matrices ∑₁ and ∑₂. The parameters μ₁, μ₂, μ₁, and ∑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.