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), 157166
MSC (2000):
Primary 62H30; Secondary 62G07
Published electronically:
September 5, 2006
MathSciNet review:
2168145
Fulltext PDF Free Access
Abstract 
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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 onedimensional 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, Teoriya raspoznavaniya obrazov. Statisticheskie
problemy obucheniya, Izdat. “Nauka”, Moscow, 1974
(Russian). MR
0474638 (57 #14274)
 2.
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
(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
(2004g:62022), http://dx.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
 1.
 V. N. Vapnik and A. Ya. Chervonenkis, Theory of Pattern Recognition, ``Nauka'', Moscow, 1974. (Russian) MR 0474638 (57:14274)
 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. Ivan'ko and R. E. Maboroda, Exponential estimates for the empirical Bayes risk in the classification of a mixture with varying concentrations, Ukr. Mat. Zh. 54 (2002), no. 10, 14211428; English transl. in Ukrainian Math. J. 54 (2002), no. 10, 17221731. MR 2015493 (2004g:62022)
 7.
 O. V. Sugakova, Asymptotics of a kernel estimate for the density of a distribution constructed from observations of a mixture with varying concentrations, Teor. Imovirnost. Matem. Stat. 59 (1998), 156166; English transl. in Theory Probab. Math. Statist. 59 (1999), 161171. 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:
http://dx.doi.org/10.1090/S0094900006006739
PII:
S 00949000(06)006739
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
