Classification of components of a mixture
Olena Sugakova · Theory of Probability and Mathematical Statistics · 2006
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.