Histogram estimates of the shape of concentration functions of two-component mixtures. (Ukrainian, English) Zbl 1124.62015

Pohyl’ko, D. I.

Teor. Jmovirn. Mat. Stat. 72, 111-119 (2005); translation in Theory Probab. Math. Stat. 72, 125-133 (2006).

The observations \(\{\xi_j,\;j=1,\dots,N\}\) are assumed to be taken from a two component mixture with varying concentrations, i.e. \[ {\mathbf P}\{\xi_j\in A\}=w(t_j)H_1(A)+(1-w(t_j))H_2(A), \] where \(H_i\) is the distribution of the \(i\)-th component of the mixture, \(w(t)\) is the concentration (mixing probability) of the first component at time \(t\), and \(t_j=j/N\) is the moment of the \(j\)-th observation. A nonparametric histogram-type estimate \(u_n\) for the shape \(u\) of \(w\) is considered, i.e., \[ u=(w-\bar w)/\| w-\bar w\| _{L_2},\quad \text{where}\quad \bar w=\int_0^1 w(t)dt. \] The rate of convergence of the estimate in \(L_2\) is investigated.

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

62G05	Nonparametric estimation
62G20	Asymptotic properties of nonparametric inference

Keywords:

mixture with varying concentrations; mixing probability