Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case

H. Richard McFarland, Donald St P. Richards

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Abstract

We consider the problem of discriminating between two independent multivariate normal populations, Np1, ∑1) and Np2, ∑2), having distinct mean vectors μ1 and μ2 and distinct covariance matrices ∑1 and ∑2. The parameters μ1, μ2, μ1, and ∑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 (Diabetologia 16 (1979), 17-24) in a classification of diabetic status.

Original languageEnglish (US)
Article number92034
Pages (from-to)299-330
Number of pages32
JournalJournal of Multivariate Analysis
Volume82
Issue number2
DOIs
Publication statusPublished - Jan 1 2002

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All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty

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