Electronic Journal of Statistics

Efficient semiparametric estimation and model selection for multidimensional mixtures

Elisabeth Gassiat, Judith Rousseau, and Elodie Vernet

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In this paper, we consider nonparametric multidimensional finite mixture models and we are interested in the semiparametric estimation of the population weights. Here, the i.i.d. observations are assumed to have at least three components which are independent given the population. We approximate the semiparametric model by projecting the conditional distributions on step functions associated to some partition. Our first main result is that if we refine the partition slowly enough, the associated sequence of maximum likelihood estimators of the weights is asymptotically efficient, and the posterior distribution of the weights, when using a Bayesian procedure, satisfies a semiparametric Bernstein-von Mises theorem. We then propose a cross-validation like method to select the partition in a finite horizon. Our second main result is that the proposed procedure satisfies an oracle inequality. Numerical experiments on simulated data illustrate our theoretical results.

Article information

Electron. J. Statist., Volume 12, Number 1 (2018), 703-740.

Received: September 2016
First available in Project Euclid: 27 February 2018

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Zentralblatt MATH identifier

Primary: 62G05: Estimation 62G20: Asymptotic properties

Semiparametric statistics mixture models efficiency Bernstein von Mises theorem

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Gassiat, Elisabeth; Rousseau, Judith; Vernet, Elodie. Efficient semiparametric estimation and model selection for multidimensional mixtures. Electron. J. Statist. 12 (2018), no. 1, 703--740. doi:10.1214/17-EJS1387. https://projecteuclid.org/euclid.ejs/1519700499

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