Recogintion of mixture components. (Ukrainian, English) Zbl 1125.62061

Sugakova, O. V.

Teor. Jmovirn. Mat. Stat. 72, 140-149 (2005); translation in Theory Probab. Math. Stat. 72, 157-166 (2006).

A classification problem is considered in the case where the teaching sample \(\{\xi_j,\;j=1,2,\dots,N\}\) is obtained from a mixture with varying concentrations, i.e., the PDF of \(\xi_j\in R^d\) is \[ f_{\xi_j}(x_1,\dots,x_d)=\sum_{k=1}^M w_j^k h_k(x_1,\dots,x_d), \] where \(h_k\) is the (unknown) PDF of the \(k\)-th class (component), and \(w_j^k\) is the (known) concentration of the \(k\)-th component at the moment of the \(j\)-th observation (mixing probability). A single-index classifier \(g_b(x)\) is considered, i.e., \(g_b(x)=g(\sum_{i=1}^d b_i x_i)\), where \(g:R^d\to\{1,\dots,M\}\) is a measurable function, and \(b_i\) are fixed coefficients of the index. The Bayesian single index classifier \(g^*_{b^*}\) minimizes the probability of misclassification through all classifiers of such kind. An empirical Bayes rule is proposed for the estimation of Bayesian \(b^*\) and \(g^*\). It is shown that the misclassification probability of the empirical Bayesian single index classifier converges to the misclassification probability of \(g^*_{b^*}\).

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

62H30	Classification and discrimination; cluster analysis (statistical aspects)
62C12	Empirical decision procedures; empirical Bayes procedures

Keywords:

single index classification; mixture with varying concentrations; consistency; empirical Bayes classifier