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December 2016 Convergence rates of parameter estimation for some weakly identifiable finite mixtures
Nhat Ho, XuanLong Nguyen
Ann. Statist. 44(6): 2726-2755 (December 2016). DOI: 10.1214/16-AOS1444


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.


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Nhat Ho. XuanLong Nguyen. "Convergence rates of parameter estimation for some weakly identifiable finite mixtures." Ann. Statist. 44 (6) 2726 - 2755, December 2016.


Received: 1 February 2015; Revised: 1 January 2016; Published: December 2016
First available in Project Euclid: 23 November 2016

zbMATH: 1359.62076
MathSciNet: MR3576559
Digital Object Identifier: 10.1214/16-AOS1444

Primary: 62F15 , 62G05
Secondary: 62G20

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

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 6 • December 2016
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