Classification of components of a mixture
Author:
O. V. Sugakova
Translated by:
S. Kvasko
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
Full-text PDF Free Access
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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 $S_j^N(b)=\sum _{i=1}^db_i \xi _j^{N,i}$ that is the projection of the vector of observations $\xi _j^N=\bigl (\xi _j^{N,1},\xi _j^{N,2}, \dots ,\xi _j^{N,d}\bigr )$ to a nonrandom direction $b=(b_1,b_2,\dots ,b_d)$. We obtain an estimator $\hat b$ for the best possible direction $b$. It is proved that the probability of error for the classifier based on $S(\hat b)$ converges to the minimal probability of error among all possible classifiers.
References
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References
- V. N. Vapnik and A. Ya. Chervonenkis, Theory of Pattern Recognition, “Nauka”, Moscow, 1974. (Russian) MR 0474638 (57:14274)
- L. Devroye and L. Györfi, Nonparametric Density Estimation, Wiley, New York, 1985. MR 0780746 (86i:62065)
- A. Afifi and S. Azen, Statistical Analysis: A Computer Oriented Approach, Academic Press, New York, 1972.
- R. E. Maĭboroda, Nonparametric Statistics of Nonhomogeneous Observations, Doctoral dissertation, Kyiv, 1994. (Ukrainian)
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- Yu. O. Ivan’ko and R. E. Maĭboroda, Exponential estimates for the empirical Bayes risk in the classification of a mixture with varying concentrations, Ukr. Mat. Zh. 54 (2002), no. 10, 1421–1428; English transl. in Ukrainian Math. J. 54 (2002), no. 10, 1722–1731. MR 2015493 (2004g:62022)
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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
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
