Adaptive moment estimates of distribution parameters by observations with admixtures. (Ukrainian, English) Zbl 1164.62332

Lodatko, N.; Majboroda, R.

Teor. Jmovirn. Mat. Stat. 75, 61-70 (2006); translation in Theory Probab. Math. Stat. 75, 71-82 (2007).

The authors consider observations with admixtures \(\Xi_{N}=\{\xi_{j:N}, j=1,\dots,N\}\), where under fixed \(N\), \(\xi_{j:N}\) are independent random variables with distribution function \[ P\{\xi_{j:N}<x\}=w_{j:N}H_1(x,\theta)+(1-w_{j:N})H_2(x), \] where \(w_{j:N}\) is the concentration of the main component in the time of the \(j\)-th observation, \(H_1(x,\theta)\) is the distribution function of main component; \(\theta\in \Theta\subset R\) is the unknown parameter; and \(H_2(x)\) is the unknown distribution function of the admixture. For the estimation of \(\theta\) by observations of \(\Xi\) the generalized method of moments is applied, where the weighted functional moments are used. The consistency and asymptotic normality of such estimates are proved. An adaptive technique, which permits to obtain estimates with the dispersion coefficient equal to the minimal value for moment estimates is proposed.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

62G05	Nonparametric estimation
62G07	Density estimation
62G20	Asymptotic properties of nonparametric inference

Keywords:

generalized method of moments; weighted functional moments