TY - JOUR

T1 - Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case

AU - McFarland, H. Richard

AU - Richards, Donald St P.

N1 - Funding Information:
1This research was supported in part by the National Science Foundation under Grants DMS-9401322 and DMS-9703705.

PY - 2001/4

Y1 - 2001/4

N2 - We consider the problem of discriminating, on the basis of random “training” samples, between two independent multivariate normal populations, Np(μ, Σ1) and Np(μ, Σ2), which have a common mean vector μ and distinct covariance matrices Σ1and Σ2Using the theory of Bessel functions of the second kind of matrix argument developed by Herz (1955, Ann. Math.61, 474-523), we derive stochastic representations for the exact distributions of the “plug-in” quadratic discriminant functions for classifying a newly obtained observation. These stochastic representations involve only chi-squared and F-distributions, hence we obtain an efficient method for simulating the discriminant functions and estimating the corresponding probabilities of misclassification. For some special values of p, Σ1and Σ2we obtain explicit formulas and inequalities for the probabilities of misclassification. We apply these results to data given by Stocks (1933, Ann. Eugen.5, 1-55) in a biometric investigation of the physical characteristics of twins, and to data provided by Rencher (1995, “Methods of Multivariate Analysis,” Wiley, New York) in a study of the relationship between football helmet design and neck injuries. For each application we estimate the exact probabilities of misclassification, and in the case of Stocks’ data we make extensive comparisons with previously published estimates.

AB - We consider the problem of discriminating, on the basis of random “training” samples, between two independent multivariate normal populations, Np(μ, Σ1) and Np(μ, Σ2), which have a common mean vector μ and distinct covariance matrices Σ1and Σ2Using the theory of Bessel functions of the second kind of matrix argument developed by Herz (1955, Ann. Math.61, 474-523), we derive stochastic representations for the exact distributions of the “plug-in” quadratic discriminant functions for classifying a newly obtained observation. These stochastic representations involve only chi-squared and F-distributions, hence we obtain an efficient method for simulating the discriminant functions and estimating the corresponding probabilities of misclassification. For some special values of p, Σ1and Σ2we obtain explicit formulas and inequalities for the probabilities of misclassification. We apply these results to data given by Stocks (1933, Ann. Eugen.5, 1-55) in a biometric investigation of the physical characteristics of twins, and to data provided by Rencher (1995, “Methods of Multivariate Analysis,” Wiley, New York) in a study of the relationship between football helmet design and neck injuries. For each application we estimate the exact probabilities of misclassification, and in the case of Stocks’ data we make extensive comparisons with previously published estimates.

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U2 - 10.1006/jmva.2000.1924

DO - 10.1006/jmva.2000.1924

M3 - Article

AN - SCOPUS:0005423688

VL - 77

SP - 21

EP - 53

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 1

ER -