Asymptotics of threshold classifiers constructed by samples from mixtures with variable concentrations. (Ukrainian, English) Zbl 1150.62383

Ivan’ko, Yu.; Maiboroda, R.

Teor. Jmovirn. Mat. Stat. 74, 34-43 (2006); translation in Theory Probab. Math. Stat. 74, 37-47 (2007).

For the item \(O\) some numerical characteristic \(\xi=\{\xi(O)\}\) is observed. This item can belong to one of two classes. This paper deals with threshold classifiers of the form \[ g_{t}=\begin{cases} 1,& \xi\leq t,\\ 2,&\xi>t,\end{cases} \] by sampling from mixtures with variable concentrations. The threshold \(t=t^{B}\) is Bayesian if \(g_{t^{B}}\) has the smallest probability of error. The authors prove that for smooth distributions the estimates obtained by the method of minimization of the empirical risk have a rate of convergence to the optimal threshold equal to \(N^{-1/3}\) (\(N\) is the sample size). It is proved that the empirical Bayes estimates give a rate of convergence of order \(N^{-2/5}\).

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

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

Keywords:

minimizing empirical risks; empirical Bayes estimates