## Maximizing estimate for parameters of concentrations of two-component mixtures.(Ukrainian, English)Zbl 1022.62019

Teor. Jmovirn. Mat. Stat. 65, 127-135 (2001); translation in Theory Probab. Math. Stat. 65, 143-152 (2002).
The authors deal with estimation of mixture component distribution functions $$H_m(x)$$ from a sample $$\xi_{1:N},\dots,\xi_{N:N}$$. The distribution function of $$\xi_{j:N}$$ is a mixture with varying concentrations $P\{\xi_{j:N}<x\}=\sum_{m=1}^2\omega^m_{j:N}(\vartheta)H_m(x),$ where $$H_m(x)$$ are distribution functions and $$\omega^m_{j:N}(\vartheta),\;\sum_{m=1}^2\omega^m_{j:N}=1,$$ 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 his previous paper, Theory Probab. Math. Stat. 64, 105-115 (2001); translation from Teor. Jmovirn. Mat. Stat. 64, 92-101 (2001; Zbl 0995.62022), the first author proposed to use the generalized least squares method to estimate $$\vartheta$$. He considered the ‘theoretical contrast function’ $J(\alpha)=\inf_{H_1,H_2}N^{-1}\sum_{j=1}^N\int_{\mathbb{R}^d}\left( P\{\xi_{j:N}<x\}-\sum_{m=1}^2\omega^m_{j:N}(\alpha)H_m(x)\right)^2\pi(dx),$ where $$\pi(\cdot)$$ is a probability measure on $$\mathbb{R}^d$$, and replaced the probability $$P\{\xi_{j:N}<x\}$$ by indicators $$I\{\xi_{j:N}<x\}$$ 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 authors propose to use a measure $$\pi(\cdot)$$ which is concentrated at a point $$x_0$$, where the function $$|H_1(x)-H_2(x)|$$ attains its maximum value. Estimates corresponding to this measure are called ‘maximizing’ estimates.

### MSC:

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

Zbl 0995.62022