Classification of components of a mixture

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.