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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II: The heterogeneous case. (English) Zbl 1099.62517
Summary: We consider the problem of discriminating between two independent multivariate normal populations, ${N}_{p}\left(\mu ,{{\Sigma }}_{1}\right)$ and ${N}_{p}\left(\mu ,{{\Sigma }}_{2}\right)$ having distinct mean vectors ${\mu }_{1}$ and ${\mu }_{2}$ and distinct covariance matrices ${{\Sigma }}_{1}$ and ${{\Sigma }}_{2}$. The parameters ${\mu }_{1}$, ${\mu }_{2}$, ${{\Sigma }}_{1}$, ${{\Sigma }}_{2}$ 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, 179–188 (1936)]; a data set on corporate financial ratios provided by R. A. Johnson and D. W. Wichern [Applied Multivariate Statistical Analysis, 4th ed., Prentice-Hall, Englewood Cliffs, NJ (1998), see Zbl 0745.62050]; and a data set analyzed by Reaven and Miller [Diabetologia 16, 17–24 (1979)] in a classification of diabetic status. For part I see J. Multivariate Anal. 77, No. 1, 21–53 (2001; Zbl 1098.62517).
##### MSC:
 62H10 Multivariate distributions of statistics 62H30 Classification and discrimination; cluster analysis (statistics) 62E15 Exact distribution theory in statistics