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.

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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