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

Majboroda, R. E.

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.

Reviewer: Mikhail Moklyachuk (Kyïv)

MSC:

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

Keywords:

varying mixtures; least squares estimates; consistency; asymptotic normality

Citations:

Zbl 0995.62022