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Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions: II. The Heterogeneous Case

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  • McFarland, H. Richard
  • Richards, Donald St. P.

Abstract

We consider the problem of discriminating between two independent multivariate normal populations, Np([mu]1, [Sigma]1) and Np([mu]2, [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, and [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 (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 (Diabetologia16 (1979), 17-24) in a classification of diabetic status.

Suggested Citation

  • McFarland, H. Richard & Richards, Donald St. P., 2002. "Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions: II. The Heterogeneous Case," Journal of Multivariate Analysis, Elsevier, vol. 82(2), pages 299-330, August.
  • Handle: RePEc:eee:jmvana:v:82:y:2002:i:2:p:299-330
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    1. McFarland, H. Richard & Richards, Donald St. P., 2001. "Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions. I. The Equal-Means Case," Journal of Multivariate Analysis, Elsevier, vol. 77(1), pages 21-53, April.
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    Cited by:

    1. Croux, Christophe & Joossens, Kristel, 2005. "Influence of observations on the misclassification probability in quadratic discriminant analysis," Journal of Multivariate Analysis, Elsevier, vol. 96(2), pages 384-403, October.

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