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Identifiability of finite mixtures of elliptical distributions. (English) Zbl 1164.62354

Finite mixture models for distributions on \(R^p\) are considered with components PDFs of the form \[ f_{p,\alpha}(x)=| \Sigma| ^{-1/2}f_0((x-\mu)^T\Sigma^{-1}(x-\mu)| \vartheta), \] \(x\in R^p\), where \(\alpha=(\vartheta,\mu,\Sigma)\) is an unknown parameter, \(f_0(\cdot| \vartheta)\) is a “density generator”, i.e., a function \([0,+\infty)\to[0,+\infty)\) such that \(\int f_0(x| \vartheta)=1\). It is shown that the mixture model \(\sum_{i=1}^m\lambda_i f_{p,\alpha_i}(x)\) is identifiable for \(p>1\) if it is identifiable for \(p=1\). Sufficient conditions of identifiability are given in terms of characteristic generators and density generators. Identifiability of mixtures of multivariate-t, multivariate symmetric stable, triangular and normal scale mixture distributions are derived as examples.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E10 Characteristic functions; other transforms
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