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Theory of Probability and Mathematical Statistics

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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
Published electronically: September 5, 2006
MathSciNet review: 2168145
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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.

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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

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

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