### Abstract

We consider the problem of discriminating between two independent multivariate normal populations, N_{p}(μ_{1}, ∑_{1}) and N_{p}(μ_{2}, ∑_{2}), having distinct mean vectors μ_{1} and μ_{2} and distinct covariance matrices ∑_{1} and ∑_{2}. The parameters μ_{1}, μ_{2}, μ_{1}, 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 language | English (US) |
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Article number | 92034 |

Pages (from-to) | 299-330 |

Number of pages | 32 |

Journal | Journal of Multivariate Analysis |

Volume | 82 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jan 1 2002 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Multivariate Analysis*,

*82*(2), 299-330. [92034]. https://doi.org/10.1006/jmva.2001.2034