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The consistency of estimators in finite mixture models. (English) Zbl 1010.62023

The classical finite mixture model is considered with i.i.d. observations in \(R^1\) and PDF \[ h(x,\alpha,\vartheta)=\sum_{i=1}^k \alpha_i f_i(x,\vartheta_i),\quad \vartheta=(\vartheta_1,\dots,\vartheta_k),\;\vartheta_i\in\Theta_i, \] where \(\Theta_i\) are closed convex sets in \(R^p\). For any given \((\alpha^0,\vartheta^0)\), \[ \Gamma(\alpha^0,\vartheta^0)=\{(\alpha,\vartheta)\;:\;h( \cdot ,\alpha,\vartheta)=h(\cdot,\alpha^0,\vartheta^0)\}. \] The authors demonstrate that the MLE for \((\vartheta,\alpha)\), say \((\hat\vartheta_n,\hat\alpha_n)\), under the usual regularity conditions is consistent in the sense that \(\text{dist}((\hat\vartheta_n,\hat\alpha_n),\Gamma(\alpha,\vartheta))\to 0\) w.p. 1, as the sample size \(n\to\infty\) (“dist” is the nearest neighbor distance). A bootstrap test is constructed to test an embedded model hypothesis (e.g., with the number of components \(k\) less than in the full model). Results of simulations are presented.

MSC:

62F12 Asymptotic properties of parametric estimators
62F40 Bootstrap, jackknife and other resampling methods
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