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Robust discrimination under a hierarchy on the scatter matrices

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  • Bianco, Ana
  • Boente, Graciela
  • Pires, Ana M.
  • Rodrigues, Isabel M.

Abstract

Under normality, Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] investigated the asymptotic properties of the quadratic discrimination procedure under hierarchical models for the scatter matrices, that is: (i) arbitrary scatter matrices, (ii) common principal components, (iii) proportional scatter matrices and (iv) identical matrices. In this paper, we study the properties of robust quadratic discrimination rules based on robust estimates of the involved parameters. Our analysis is based on the partial influence functions of the functionals related to these parameters and allows to derive the asymptotic variances of the estimated coefficients under models (i)-(iv). From them, we conclude that the asymptotic variances verify the same order relations as those obtained by Flury and Schmid [Quadratic discriminant functions with constraints on the covariances matrices: some asymptotic results, J. Multivariate Anal. 40 (1992) 244-261] for the classical estimators. We also perform a Monte Carlo study for different sample sizes and different hierarchies which shows the advantage of using robust procedures over classical ones, when anomalous data are present. It also confirms that better rates of misclassification can be achieved if a more parsimonious model among all the correct ones is used instead of the standard quadratic discrimination.

Suggested Citation

  • Bianco, Ana & Boente, Graciela & Pires, Ana M. & Rodrigues, Isabel M., 2008. "Robust discrimination under a hierarchy on the scatter matrices," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1332-1357, July.
    • Handle: RePEc:eee:jmvana:v:99:y:2008:i:6:p:1332-1357
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    References listed on IDEAS

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    1. Bernhard Flury & Martin Schmid & A. Narayanan, 1994. "Error rates in quadratic discrimination with constraints on the covariance matrices," Journal of Classification, Springer;The Classification Society, vol. 11(1), pages 101-120, March.
    2. Boente, Graciela & Pires, Ana M. & Rodrigues, Isabel M., 2006. "General projection-pursuit estimators for the common principal components model: influence functions and Monte Carlo study," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 124-147, January.
    3. 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.
    4. Pires, Ana M. & Branco, João A., 2002. "Partial Influence Functions," Journal of Multivariate Analysis, Elsevier, vol. 83(2), pages 451-468, November.
    5. Flury, Bernhard W. & Schmid, Martin J., 1992. "Quadratic discriminant functions with constraints on the covariance matrices: Some asymptotic results," Journal of Multivariate Analysis, Elsevier, vol. 40(2), pages 244-261, February.
    6. Norm A. Campbell, 1978. "The Influence Function as an Aid in Outlier Detection in Discriminant Analysis," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 27(3), pages 251-258, November.
    7. Gervini, Daniel, 2002. "The influence function of the Stahel-Donoho estimator of multivariate location and scatter," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 425-435, December.
    8. Graciela Boente, 2002. "Influence functions and outlier detection under the common principal components model: A robust approach," Biometrika, Biometrika Trust, vol. 89(4), pages 861-875, December.
    9. Graciela Boente & Frank Critchley & Liliana Orellana, 2007. "Influence functions of two families of robust estimators under proportional scatter matrices," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 15(3), pages 295-327, February.
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    Cited by:

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