Estimation of the density of a distribution from data with an admixture. (Ukrainian, English) Zbl 1114.62042

Lodatko, N.; Maiboroda, R.

Teor. Jmovirn. Mat. Stat. 73, 88-96 (2005); translation in Theory Probab. Math. Stat., Vol 73, 99-108 (2006).

The authors consider the problem of estimation of the density function from observations of two-component mixtures with varying concentrations. Let \(\xi_{1:N},\dots,\xi_{N:N}\) be a sample from a mixture with varying concentrations of independent random variables with the distribution function \[ P\{\xi_{j:N}<x\}=\omega_{j:N}H_1(x)+(1-\omega_{j:N})H_2(x), \] where \(H_1(x)\) is the distribution function of the basic component, \(H_2(x)\) is the distribution function of the admixture component, and \(\omega_{j:N}\) is the concentration of the basic component at the moment of the \(j\)-th observation. It is assumed that the distribution of the first component is unknown, while a parametric model is (perhaps) available for the second component.
Applying the sieve maximum likelihood method the authors construct histogram-type estimators \(\tilde{h}_i^N(x)\) for the densities \(h_i(x),i=1,2,\) of distributions of the components in the class of functions \(g_k(x)=\sum_{k=1}^{K_N}g_k{\mathbb I}\{x\in A_k\},\) where \(K_N\) is the amount of subintervals of the partition \(A_k\), \[ A_k=[t_{k-1},t_k), \quad k=1,2,\dots,K_n-1,\qquad A_{K_N}= [t_{K_N-1},t_{K_N}], \quad t_k=k/K_N, \] \(g_i\) are positive numbers such that \(K_n^{-1}\sum_{k=1}^{K_N}g_k=1\). They prove the consistency of the estimators and obtain estimates for the rate of convergence.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

62G07	Density estimation
62G20	Asymptotic properties of nonparametric inference

Keywords:

sieve maximum likelihood method; histogram; mixture with varying concentrations; consistency