×

Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I: The equal-means case. (English) Zbl 1098.62530

Summary: We consider the problem of discriminating, on the basis of random ‘training’ samples, between two independent multivariate normal populations, \(N_p(\mu, \Sigma_1)\) and \(N_p(\mu, \Sigma_2)\), which have a common mean vector \(\mu\) and distinct covariance matrices \(\Sigma_1\) and \(\Sigma_2\). Using the theory of Bessel functions of the second kind of matrix argument developed by C. S. Herz [Ann. Math. 61, 474–523 (1955; Zbl 0066.32002)], 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, \(\Sigma_1\) and \(\Sigma_2\) we obtain explicit formulas and inequalities for the probabilities of misclassification. We apply these results to data given by Stocks [Ann. Eugen. 5, 1–55 (1933)] in a biometric investigation of the physical characteristics of twins, and to data provided by A. C. Rencher [Methods of Multivariate Analysis. (1995; Zbl 0836.62039)] 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.

MSC:

62H10 Multivariate distribution of statistics
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62E15 Exact distribution theory in statistics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd ed., Wiley, New York.; T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd ed., Wiley, New York. · Zbl 0083.14601
[2] Bartlett, M. S.; Please, N. W., Discrimination in the case of zero mean differences, Biometrika, 50, 17-21 (1963) · Zbl 0136.39401
[3] Bowker, A. H., A representation of Hotelling’s \(T^2\) and Anderson’s classification statistic \(W\) in terms of simple statistics, (Solomon, H., Studies in Item Analysis and Prediction (1961), Stanford Univ. Press: Stanford Univ. Press Stanford), 285-292 · Zbl 0146.40302
[4] Desu, M.; Geisser, S., Methods and applications of equal-mean discrimination, (Cacoullos, T., Discriminant Analysis and Applications (1970), Academic Press: Academic Press New York), 139-159
[5] Gabriel, K. R., The biplot graphic display of matrices with application to principal component analysis, Biometrika, 58, 453-467 (1971) · Zbl 0228.62034
[6] Herz, C. S., Bessel functions of matrix argument, Ann. of Math., 61, 474-523 (1955) · Zbl 0066.32002
[7] Johnson, N. L.; Kotz, S., Continuous Univariate Distributions-2 (1970), Houghton-Mifflin: Houghton-Mifflin Boston · Zbl 0213.21101
[8] Johnson, R. A.; Wichern, D. W., Applied Multivariate Statistical Analysis (1992), Prentice-Hall: Prentice-Hall NJ · Zbl 0745.62050
[9] Khattree, R.; Naik, D. N., Applied Multivariate Statistics with \(SAS^®\) Software (1996), SAS Institute: SAS Institute Cary
[10] Marco, V. R.; Young, D. M.; Turner, D. W., Asymptotic expansions and estimation of the expected error rate for equal-mean discrimination with uniform covariance structure, Biometrical J., 29, 103-113 (1987) · Zbl 0613.62080
[11] Mardia, K. V., Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies, Sankhyā, Ser. B, 36, 115-128 (1974) · Zbl 0345.62031
[12] McFarland, H. R., The Exact Distributions of ‘Plug-in’ Discriminant Functions in Multivariate Analysis (1998), University of Virginia
[13] McLachlan, G. J., Discriminant Analysis and Statistical Pattern Recognition (1992), Wiley: Wiley New York
[14] Muirhead, R. J., Latent roots and matrix variates: a review of some asymptotics, Ann. Statist., 6, 5-33 (1978) · Zbl 0375.62050
[15] Muirhead, R. J., Aspects of Multivariate Statistical Theory (1982), Wiley: Wiley New York · Zbl 0556.62028
[16] Okamoto, M., Discrimination for variance matrices, Osaka Math. J., 13, 1-39 (1961) · Zbl 0098.32903
[17] Rencher, A. C., Methods of Multivariate Analysis (1995), Wiley: Wiley New York · Zbl 0836.62039
[18] Siotani, M., Large sample approximations and asymptotic expansions of classification statistics, (Krishnaiah, P. R.; Kanal, L. N., Classification, Pattern Recognition and Reduction of Dimensionality, Handbook of Statistics (1984), North-Holland: North-Holland New York), 61-100
[19] Stocks, P., A biometric investigation of twins, Part II, Ann. Eugen., 5, 1-55 (1933)
[20] Watson, G. N., A Treatise on the Theory of Bessel Functions (1922), Cambridge Univ. Press: Cambridge Univ. Press New York · JFM 48.0412.02
[21] Young, D. M.; Seaman, J. W.; Jennings, L. W., Bounds on the error rate for equal-mean discrimination with known population parameters, Statistica, 47, 85-96 (1987) · Zbl 0635.62052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.