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 language | English |
|---|---|
| Pages (from-to) | 300-326 |
| Number of pages | 27 |
| Journal | Sankhya: The Indian Journal of Statistics |
| Volume | 75 A |
| Issue number | PART2 |
| Publication status | Published - 2013 |
| Externally published | Yes |
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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
Cite this
On Kolmogorov asymptotics of estimators of the misclassification error rate in linear discriminant analysis. / Zollanvari, Amin; Genton, Marc G.
In: Sankhya: The Indian Journal of Statistics, Vol. 75 A, No. PART2, 2013, p. 300-326.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - On Kolmogorov asymptotics of estimators of the misclassification error rate in linear discriminant analysis
AU - Zollanvari, Amin
AU - Genton, Marc G.
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
KW - Double asymptotics
KW - Error estimation
KW - Kolmogorov asymptotic analysis
KW - Plug-in error
KW - Resubstitution
KW - Smoothed resubstitution
KW - True error
UR - http://www.scopus.com/inward/record.url?scp=84881576929&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84881576929&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:84881576929
VL - 75 A
SP - 300
EP - 326
JO - Sankhya: The Indian Journal of Statistics
JF - Sankhya: The Indian Journal of Statistics
SN - 0972-7671
IS - PART2
ER -