The asymptotic behavior of threshold-based classification rules constructed from a sample from a mixture with varying concentrations
Authors:
Yu. Ivan’ko and R. Maĭboroda
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 74 (2007), 37-47
MSC (2000):
Primary 62H30; Secondary 62G07
DOI:
https://doi.org/10.1090/S0094-9000-07-00696-5
Published electronically:
June 25, 2007
MathSciNet review:
2336777
Full-text PDF Free Access
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Additional Information
Abstract: We consider a problem on finding the best threshold-based classification rule constructed from a sample from a mixture with varying concentrations. We show that the rate of convergence of the minimal empirical risk estimators to the optimal threshold is of order $N^{-1/3}$ for smooth distributions, while the rate of convergence of the Bayes empirical estimators is of order $N^{-2/5}$ where $N$ is the size of a sample.
References
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References
- P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York, 1968. MR 0233396 (38:1718)
- V. N. Vapnik, Inductive principles for the search for empirical laws, Pattern Recognition. Classification. Prediction, vol. 1, “Nauka”, Moscow, 1989, pp. 17–81. (Russian) MR 1074215 (92e:68170)
- L. Devroye and L. Gyorfi, Nonparametric Density Estimation. The $L_1$ View, John Wiley & Sons, Inc., New York, 1985. MR 780746 (86i:62065)
- R. E. Maĭboroda, Statistical Analysis of Mixtures. A course of lectures, “Kyiv University”, Kyiv, 2003. (Ukrainian)
- Yu. O. Ivan’ko and R. E. Maĭboroda, Exponential estimates for the empirical Bayes risk in the classification of a mixture with varying concentrations, Ukrain. Mat. Zh. 54 (2002), no. 10, 1421–1428; English transl. in Ukrainian Math. J. 54 (2002), no. 10, 1722–1731. MR 2015493 (2004g:62022)
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Additional Information
Yu. Ivan’ko
Affiliation:
SK Lemma-Vite, Brats’ka Street, Kyiv, 6, 04070 Ukraine
Email:
ivanko@lemma-insur.com.ua
R. Maĭboroda
Affiliation:
Department of Probability Theory and Mathematical Statistics, Faculty for Mathematics and Mechanics, National Taras Shevchenko University, Glushkov Avenue, 6, Kyiv, 03127, Ukraine
Email:
mre@univ.kiev.ua
Keywords:
Minimization of the empirical risk,
kernel estimators of densities,
Bayes empirical classification rule,
estimates of components of a mixture,
mixtures with varying concentrations
Received by editor(s):
December 20, 2004
Published electronically:
June 25, 2007
Article copyright:
© Copyright 2007
American Mathematical Society
