On Kolmogorov asymptotics of estimators of the misclassification error rate in linear discriminant analysis

Amin Zollanvari, Marc G. Genton

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

Abstract

We provide a fundamental theorem that can be used in conjunction with Kolmogorov asymptotic conditions to derive the first moments of well-known estimators of the actual error rate in linear discriminant analysis of a multi-variate Gaussian model under the assumption of a common known covariance matrix. The estimators studied in this paper are plug-in and smoothed resub-stitution error estimators, both of which have not been studied before under Kolmogorov asymptotic conditions. As a result of this work, we present an optimal smoothing parameter that makes the smoothed resubstitution an unbiased estimator of the true error. For the sake of completeness, we further show how to utilize the presented fundamental theorem to achieve several previously reported results, namely the first moment of the resubsti-tution estimator and the actual error rate. We provide numerical examples to show the accuracy of the succeeding finite sample approximations in situations where the number of dimensions is comparable or even larger than the sample size.

Original languageEnglish
Pages (from-to)300-326
Number of pages27
JournalSankhya: The Indian Journal of Statistics
Volume75 A
Issue numberPART2
Publication statusPublished - Aug 21 2013

Keywords

  • Double asymptotics
  • Error estimation
  • Kolmogorov asymptotic analysis
  • Plug-in error
  • Resubstitution
  • Smoothed resubstitution
  • True error

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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