Asymptotic of empirical Bayes risk for the classification by a mixture of two components with varying concentrations. (Ukrainian, English) Zbl 1071.62056

Ivan’ko, Yu. O.

Teor. Jmovirn. Mat. Stat. 70, 47-53 (2004); translation in Theory Probab. Math. Stat. 70, 53-60 (2005).

An empirical Bayes classifier is described for the case where the teaching sample \(\Xi_N\) is taken from a mixture with varying concentrations, i.e., \(\Xi_N=(\xi_1,\dots,\xi_N)\), where \(\xi_j\) are independent random variables and \[ \Pr\{\xi_j<x\}=w_j H_1(x)+(1-w_j)H_2(x), \] \(H_i\) is the distribution of the unknown \(i\)-th component (class) in the mixture and \(w_j\) are known mixing probabilities which vary from observation to observation. The classifier is based on kernel estimates of the PDF for both classes. Let \(L^*\) be the probability of error of the true Bayesian classifier and let \(L_N\) be the probability (conditioned by \(\Xi_N\)) of error of the constructed empirical Bayes classifier. It is shown that for smooth enough \(H_i\) \[ N^{4/5}(L_N-L^*)\Rightarrow \zeta^2, \] where \(A,B\in R\), \(\zeta\sim N(A,B^2)\).

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

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

Keywords:

kernel density estimation; empirical Bayes classification; asymptotic distribution