Convergence rates of parameter estimation for some weakly identifiable finite mixtures
The Annals of Statistics · 2016 · Volume 44 · Issue 6
Abstract
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.
Keywords
62F15, 62G05, 62G20, maximum likelihood estimation, minimax bounds, Mixture models, strong identifiability, system of polynomial equations, Wasserstein distances, weak identifiability
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