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Asymptotic Expansion and Estimation of EPMC for Linear Classification Rules in High Dimension

Author

Listed:
  • Tatsuya Kubokawa

    (Faculty of Economics, University of Tokyo)

  • Masashi Hyodo

    (Graduate School of Economics, University of Tokyo)

  • Muni S. Srivastava

    (Department of Statistics, University of Toronto,)

Abstract

The problem of classifying a new observation vector into one of the two known groups distributed as multivariate normal with common covariance matrix is consid- ered. In this paper, we handle the situation that the dimension, p, of the observation vectors is less than the total number, N, of observation vectors from the two groups, but both p and N tend to in nity with the same order. Since the inverse of the sample covariance matrix is close to an ill condition in this situation, it may be better to replace it with the inverse of the ridge-type estimator of the covariance matrix in the linear discriminant analysis (LDA). The resulting rule is called the ridge-type linear discriminant analysis (RLDA). The second-order expansion of the expected probability of misclassi cation (EPMC) for RLDA is derived, and the second-order unbiased estimator of EMPC is given. These results not only provide the corresponding conclusions for LDA, but also clarify the condition that RLDA improves on LDA in terms of EPMC. Finally, the performances of the second-order approximation and the unbiased estimator are investigated by simulation.

Suggested Citation

  • Tatsuya Kubokawa & Masashi Hyodo & Muni S. Srivastava, 2011. "Asymptotic Expansion and Estimation of EPMC for Linear Classification Rules in High Dimension," CIRJE F-Series CIRJE-F-818, CIRJE, Faculty of Economics, University of Tokyo.
    • Handle: RePEc:tky:fseres:2011cf818
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    References listed on IDEAS

    as
    1. Fujikoshi, Yasunori, 2000. "Error Bounds for Asymptotic Approximations of the Linear Discriminant Function When the Sample Sizes and Dimensionality are Large," Journal of Multivariate Analysis, Elsevier, vol. 73(1), pages 1-17, April.
    2. Saranadasa, H., 1993. "Asymptotic Expansion of the Misclassification Probabilities of D- and A-Criteria for Discrimination from Two High Dimensional Populations Using the Theory of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 46(1), pages 154-174, July.
    3. Kubokawa, Tatsuya & Srivastava, Muni S., 2008. "Estimation of the precision matrix of a singular Wishart distribution and its application in high-dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 99(9), pages 1906-1928, October.
    4. Srivastava, Muni S., 2006. "Minimum distance classification rules for high dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 97(9), pages 2057-2070, October.
    5. Gérard Letac & Hélène Massam, 2004. "All Invariant Moments of the Wishart Distribution," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(2), pages 295-318, June.
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    Cited by:

    1. Tatsuya Kubokawa & Akira Inoue, 2012. "Estimation of Covariance and Precision Matrices in High Dimension," CIRJE F-Series CIRJE-F-855, CIRJE, Faculty of Economics, University of Tokyo.

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