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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, $N_p(\mu, \Sigma_1)$ and $N_p(\mu, \Sigma_2)$ having distinct mean vectors $\mu_1$ and $\mu_2$ and distinct covariance matrices $\Sigma_1$ and $\Sigma_2$. The parameters $\mu_1$, $\mu_2$, $\Sigma_1$, $\Sigma_2$ 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 {\it R. A. Johnson} and {\it 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:
62H10Multivariate distributions of statistics
62H30Classification and discrimination; cluster analysis (statistics)
62E15Exact distribution theory in statistics
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References:
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