Classification of components of a mixture

Author:
O. V. Sugakova

Translated by:
S. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **72** (2005).

Journal:
Theor. Probability and Math. Statist. **72** (2006), 157-166

MSC (2000):
Primary 62H30; Secondary 62G07

DOI:
https://doi.org/10.1090/S0094-9000-06-00673-9

Published electronically:
September 5, 2006

MathSciNet review:
2168145

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of classification of individuals sampled from a mixture of several components with different probability distributions. To construct a classifier we use kernel estimators of the density of components in the mixture for a one-dimensional random variable that is the projection of the vector of observations to a nonrandom direction . We obtain an estimator for the best possible direction . It is proved that the probability of error for the classifier based on converges to the minimal probability of error among all possible classifiers.

**1.**V. N. Vapnik and A. Ya. Chervonenkis,*\cyr Teoriya raspoznavaniya obrazov. Statisticheskie problemy obucheniya.*, Izdat. “Nauka”, Moscow, 1974 (Russian). MR**0474638****2.**L. Devroye and L. Györfi,*Nonparametric Density Estimation*, Wiley, New York, 1985. MR**0780746 (86i:62065)****3.**A. Afifi and S. Azen,*Statistical Analysis: A Computer Oriented Approach*, Academic Press, New York, 1972.**4.**R. E. Maboroda,*Nonparametric Statistics of Nonhomogeneous Observations*, Doctoral dissertation, Kyiv, 1994. (Ukrainian)**5.**-,*Statistical Analysis of Mixtures. A Course of Lectures*, Kyiv University, Kyiv, 2004. (Ukrainian)**6.**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**, https://doi.org/10.1023/A:1023792522291**7.**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**

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Additional Information

**O. V. Sugakova**

Affiliation:
Department of Mathematics and Theoretical Radiophysics, Faculty for Radiophysics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Email:
sugak@univ.kiev.ua

DOI:
https://doi.org/10.1090/S0094-9000-06-00673-9

Keywords:
Kernel estimators of the density,
Bayes empirical classifier,
estimates of components of a mixture

Received by editor(s):
April 2, 2004

Published electronically:
September 5, 2006

Article copyright:
© Copyright 2006
American Mathematical Society