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.


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