## Least squares estimate for parameters of concentrations of varying mixtures. II: Asymptotic normality.(Ukrainian, English)Zbl 1023.62023

Teor. Jmovirn. Mat. Stat. 65, 121-126 (2001); translation in Theory Probab. Math. Stat. 65, 135-142 (2002).
This is the second part of the author’s article, Theory Probab. Math. Stat. 64, 105-115 (2001); translation from Teor. Jmovirn. Mat. Stat. 64, 92-101 (2001; Zbl 0995.62022). The author deals with estimation of mixture components distributions $$H_m(A)$$ from a sample $$\xi_{1:N},\dots,\xi_{N:N}$$. The distribution of $$\xi_{j:N}$$ is a mixture with varying concentrations $P\{\xi_{j:N}\in A\}=\sum_{m=1}^M\omega^m_{j:N}(\vartheta)H_m(A),$ where $$\omega^m_{j:N}(\vartheta),\;\sum_{m=1}^M\omega^m_{j:N}=1,$$ are real numbers called concentrations of the $$m$$-th component at the moment of the $$j$$-th observation. Concentrations $$\omega^m_{j:N}(\vartheta)$$ are supposed to be known up to the unknown parameter $$\vartheta$$. In part I the author proposed to use the generalized least squares method to estimate $$\vartheta$$. He considered the ‘theoretical contrast function’ $J(\alpha)= \min_{G_m} N^{-1} \sum_{j=1}^N \int_S\left( P\{\xi_{j:N}\in A\}- \sum_{m=1}^M \omega^m_{j:N}(\alpha) G_m(A)\right)^2\pi(dx),$ where $$\pi(\cdot)$$ is a probability measure on $$S$$, and replaced the probability $$P\{\xi_{j:N}\in A\}$$ by indicators $$I\{\xi_{j:N}\in A\}$$ to get the empirical contrast function $$J_N(\alpha)$$. A point of minimum of the empirical contrast function $$J_N(\alpha)$$ is considered as an estimate of $$\vartheta$$. Consistency of this estimate was proved. In the present paper the asymptotic normality of the proposed estimates is proved as $$N\to\infty$$. An example of application of the general results is proposed.

### MSC:

 62F12 Asymptotic properties of parametric estimators 62G20 Asymptotic properties of nonparametric inference 62G05 Nonparametric estimation 62F10 Point estimation

Zbl 0995.62022