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A Bhattacharyya-type Conditional Error Bound for Quadratic Discriminant Analysis

Author

Listed:
  • Ata Kabán

    (University of Birmingham)

  • Efstratios Palias

    (University of Birmingham)

Abstract

We give an upper bound on the conditional error of Quadratic Discriminant Analysis (QDA), conditioned on parameter estimates. In the case of maximum likelihood estimation (MLE), our bound recovers the well-known Chernoff and Bhattacharyya bounds in the infinite sample limit. We perform an empirical assessment of the behaviour of our bound in a finite sample MLE setting, demonstrating good agreement with the out-of-sample error, in contrast with the simpler but uninformative estimated error, which exhibits unnatural behaviour with respect to the sample size. Furthermore, our conditional error bound is applicable whenever the QDA decision function employs parameter estimates that differ from the true parameters, including regularised QDA.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:26:y:2024:i:4:d:10.1007_s11009-024-10105-x
    DOI: 10.1007/s11009-024-10105-x
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    References listed on IDEAS

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    1. 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.
    2. Jianqing Fan & Yuan Liao & Han Liu, 2016. "An overview of the estimation of large covariance and precision matrices," Econometrics Journal, Royal Economic Society, vol. 19(1), pages 1-32, February.
    3. Schäfer Juliane & Strimmer Korbinian, 2005. "A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 4(1), pages 1-32, November.
    4. 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.
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