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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

Citations:

Zbl 0995.62022
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