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

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.

**1.**O. V. Sugakova,*Asymptotics of a kernel estimate for the density of a distribution constructed from observations of a mixture with varying concentration*, Teor. Ĭmovīr. Mat. Stat.**59**(1998), 156–166 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**59**(1999), 161–171 (2000). MR**1793776****2.**R. Ē. Maĭboroda,*Estimation of the distributions of the components of mixtures having varying concentrations*, Ukraïn. Mat. Zh.**48**(1996), no. 4, 558–562 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J.**48**(1996), no. 4, 618–622 (1997). MR**1417019**, 10.1007/BF02390622**3.**Luc Devroye and László Györfi,*Nonparametric density estimation*, Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics, John Wiley & Sons, Inc., New York, 1985. The 𝐿₁ view. MR**780746****4.**Yu. O. Īvan′ko and R. Ē. Maĭboroda,*Exponential estimates for the empirical Bayes risk in the classification of a mixture with varying concentrations*, Ukraïn. Mat. Zh.**54**(2002), no. 10, 1421–1428 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J.**54**(2002), no. 10, 1722–1731. MR**2015493**, 10.1023/A:1023792522291**5.**Yu. O. Ivan'ko,*Asymptotic behavior of kernel estimators of densities and their derivatives constructed from samples with varying concentrations*, Visnyk Kyiv. Univer.**9**(2003), 29-35. (Ukrainian)**6.**R. E. Maboroda,*Statistical Analysis of Mixtures*, Kyiv University, Kyiv, 2003. (Ukrainian)**7.**A. A. Borovkov,*Teoriya veroyatnostei*, 2nd ed., “Nauka”, Moscow, 1986 (Russian). MR**891184**

A. A. Borovkov,*Probability theory*, Gordon and Breach Science Publishers, Amsterdam, 1998. Translated from the 1986 Russian original by O. Borovkova and revised by the author. MR**1711261****8.**V. V. Anisimov and A. V. Seilhamer,*Asymptotic properties of extreme sets of random fields*, Teor. Ĭmovīr. Mat. Stat.**51**(1994), 30–38 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**51**(1995), 29–38 (1996). MR**1445049**

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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