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

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 $\Re =[a,b] \subset \mathbf R$ and smooth distribution densities, we find the precise rate of convergence for the error $L_N$ of the empirical Bayes classifier $g_N$ to the error $L^\ast$ of the Bayes classifier, namely we prove that \[ N^{4 / 5}(L_N-L^\ast ) \Rightarrow [A+B\varsigma ]^2 \] where $\varsigma$ is a standard normal random variable, and the empirical Bayes classifier $g_N$ 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.