The Annals of Statistics

Convergence rates of parameter estimation for some weakly identifiable finite mixtures

Nhat Ho and XuanLong Nguyen

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We establish minimax lower bounds and maximum likelihood convergence rates of parameter estimation for mean-covariance multivariate Gaussian mixtures, shape-rate Gamma mixtures and some variants of finite mixture models, including the setting where the number of mixing components is bounded but unknown. These models belong to what we call “weakly identifiable” classes, which exhibit specific interactions among mixing parameters driven by the algebraic structures of the class of kernel densities and their partial derivatives. Accordingly, both the minimax bounds and the maximum likelihood parameter estimation rates in these models, obtained under some compactness conditions on the parameter space, are shown to be typically much slower than the usual $n^{-1/2}$ or $n^{-1/4}$ rates of convergence.

Article information

Ann. Statist., Volume 44, Number 6 (2016), 2726-2755.

Received: February 2015
Revised: January 2016
First available in Project Euclid: 23 November 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F15: Bayesian inference 62G05: Estimation
Secondary: 62G20: Asymptotic properties

Mixture models strong identifiability weak identifiability Wasserstein distances minimax bounds maximum likelihood estimation system of polynomial equations


Ho, Nhat; Nguyen, XuanLong. Convergence rates of parameter estimation for some weakly identifiable finite mixtures. Ann. Statist. 44 (2016), no. 6, 2726--2755. doi:10.1214/16-AOS1444.

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Supplemental materials

  • Supplement to “Convergence rates of parameter estimation for some weakly identifiable finite mixtures”. In this supplemental material, we present the proofs of several remaining technical lemmas.