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Theory of Probability and Mathematical Statistics

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Asymptotics of empirical Bayes risk in the classification of a mixture of two components with varying concentrations


Author: Yu. O. Ivan'ko
Translated by: Oleg Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 70 (2004).
Journal: Theor. Probability and Math. Statist. 70 (2005), 53-60
MSC (2000): Primary 62H30; Secondary 62C10, 62C12
DOI: https://doi.org/10.1090/S0094-9000-05-00630-7
Published electronically: August 26, 2005
MathSciNet review: 2109822
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of classification for a sample from a mixture of several components. For the problem of classification of a two-component mixture with the space of characteristics $\Re=[a,b] \subset\mathbf R$and smooth distribution densities, we find the precise rate of convergence for the error $L_N$ of the empirical Bayes classifier $g_N$ to the error $L^\ast$ of the Bayes classifier, namely we prove that

\begin{displaymath}N^{4 / 5}(L_N-L^\ast) \Rightarrow [A+B\varsigma ]^2 \end{displaymath}

where $\varsigma$ is a standard normal random variable, and the empirical Bayes classifier $g_N$ is constructed from the kernel estimator of the density of a mixture with varying concentrations. We prove that the kernel estimator with the Epanechnikov kernel is optimal for the empirical Bayes classifier.


References [Enhancements On Off] (What's this?)

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Additional Information

Yu. O. Ivan'ko
Affiliation: Department of Probability Theory and Mathematical Statistics, Kyiv National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: ivanko@lemma-insur.com.ua

DOI: https://doi.org/10.1090/S0094-9000-05-00630-7
Received by editor(s): April 4, 2003
Published electronically: August 26, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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