IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v77y2001i1p21-53.html
   My bibliography  Save this article

Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions. I. The Equal-Means Case

Author

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

Abstract

We consider the problem of discriminating, on the basis of random "training" samples, between two independent multivariate normal populations, Np([mu], [Sigma]1) and Np([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 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, [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 (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.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:jmvana:v:77:y:2001:i:1:p:21-53
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(00)91924-9
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. Ata Kabán & Efstratios Palias, 2024. "A Bhattacharyya-type Conditional Error Bound for Quadratic Discriminant Analysis," Methodology and Computing in Applied Probability, Springer, vol. 26(4), pages 1-17, December.
    3. Dário Ferreira & Sandra S. Ferreira, 2024. "Optimizing Allocation Rules in Discrete and Continuous Discriminant Analysis," Mathematics, MDPI, vol. 12(8), pages 1-15, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:77:y:2001:i:1:p:21-53. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.