## 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

### Keywords:

embedded model; maximum likelihood; parametric bootstrap
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