## 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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### References:

 [1] Abramowitz M., Handbook of mathematical functions, with formulas, graphs, and mathematical tables (1965) · Zbl 0171.38503 [2] DOI: 10.1109/TIT.1981.1056389 · Zbl 0465.60024 [3] Bohning D., Computer-assisted analysis of mixtures and applications: meta-analysis, disease mapping and others (2000) [4] Box G. E. P., Bayesian inference in statistical analysis (1992) · Zbl 0850.62004 [5] DOI: 10.1016/0047-259X(81)90082-8 · Zbl 0469.60019 [6] Chandra S., Scand. J. Statist. 4 pp 105– (1977) [7] Fang K.-T., Symmetric multivariate and related distributions (1990) [8] Feller W., An introduction to probability theory and its applications (1971) · Zbl 0219.60003 [9] DOI: 10.1198/016214502760047131 · Zbl 1073.62545 [10] Gneiting T., J. Statist. Comput. Simul. 59 pp 375– (1997) [11] DOI: 10.2307/2283852 · Zbl 0155.27204 [12] Holzmann H., Sankhya 66 pp 440– (2004) [13] Holzmann H., Biometrics (2006) [14] Kay S., Modern spectral estimation: theory and applications (1988) · Zbl 0658.62108 [15] Kelker D., Sankhya Ser. A 32 pp 419– (1970) [16] Kent J. T., Ann. Statist. 11 pp 984– (1983) [17] Kotz S., Multivariate t distributions and their applications (2004) · Zbl 1100.62059 [18] Leroux B. G., Ann. Statist. 20 pp 1350– (1992) [19] Lindsay B. G., Mixture models: theory, geometry, and applications (1995) · Zbl 1163.62326 [20] DOI: 10.1214/088342304000000404 · Zbl 1100.62516 [21] DOI: 10.1111/j.0006-341X.2003.00129.x · Zbl 1274.62821 [22] McLachlan G. J., Finite mixture models (2000) · Zbl 0963.62061 [23] DOI: 10.1023/A:1008981510081 [24] Teicher H., Ann. Math. Statist. 32 pp 244– (1961) [25] Teicher H., Ann. Math. Statist. 34 pp 1265– (1963) [26] Titterington D. M., Statistical analysis of finite mixture models (1985) · Zbl 0646.62013 [27] Watson G. N., A treatise on the theory of bessel functions (1944) · Zbl 0063.08184 [28] Yakowitz S. J., Ann. Math. Statist. 39 pp 209– (1968)
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