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

Full-text PDF Free Access

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 and smooth distribution densities, we find the precise rate of convergence for the error of the empirical Bayes classifier to the error of the Bayes classifier, namely we prove that

where is a standard normal random variable, and the empirical Bayes classifier 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.

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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