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We consider the problem of classification for a sample from a mixture of several components. For the problem of classification of a two-component mixture with the space of characteristics = [a, b] ⊂ R and smooth distribution densities, we find the precise rate of convergence for the error LN of the empirical Bayes classifier gN to the error L ∗ of the Bayes classifier, namely we prove that N(LN − L∗) ⇒ [A + Bς] where ς is a standard normal random variable, and the empirical Bayes classifier gN is constructed from the kernel estimator of the density of a mixture with varying concentrations. We prove that the kernel estimator with the Epanechnikov kernel is optimal for the empirical Bayes classifier.

Asymptotics of empirical Bayes risk in the classification of a mixture of two components with varying concentrations