Strong identifiability and optimal minimax rates for finite mixture estimation
type: Item
instance of: scholarly article
title: Strong identifiability and optimal minimax rates for finite mixture estimation
zbMATH Open document ID: 1420.62215
DOI: 10.1214/17-AOS1641
author: Philippe Heinrich
author: Jonas Kahn
published in: The Annals of Statistics
publication date: 2018-10-30T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.aos/1536307235
review text: Excerpt from abstract and paper: ``We study the rates of estimation of finite mixing distributions, that is, the parameters of the mixture. We prove that under some regularity and strong identifiability conditions, around a given mixing distribution with $m_0$ components, the optimal local minimax rate of estimation of a mixing distribution with $m$ components is $n^{-1/(4(m-m_0)+2)}$. This corrects a previous paper by \textit{J. Chen} [Ann. Stat. 23, No. 1, 221--233 (1995; Zbl 0821.62023)]. [\dots] The rate gets worse with more components, which is consistent with the behaviour when there are infinitely many components, such as deconvolution. [...] In addition, the optimal local minimax rate and the optimal pointwise rate of estimation everywhere are not the same. This discrepancy is unusual in statistics, and probably the reason why the $n^{-1/4}$ rate went unchallenged for twenty years. Specifically, if instead of comparing all pairs of mixtures in a ball, we allow only one mixture in it, we get (21) which corrects Lemma 2 of Chen [loc. cit.]. As a consequence, Theorem 2 of Chen is valid by dropping uniformity: for any fixed mixing distribution say $G$, the estimator considered there will converge at rate $n^{-1/4}$, but with a multiplicative constant that depends on $G$. It then becomes a statement on the optimal pointwise rate of estimation everywhere, and can even be strengthened to $n^{-1/2}$.''
reviewed by: Claudia Kirch
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 62H30
zbMATH DE Number: 6968601
zbMATH Keywords: local asymptotic normality
zbMATH Keywords: convergence of experiments
zbMATH Keywords: maximum likelihood estimate
zbMATH Keywords: Wasserstein metric
zbMATH Keywords: mixing distribution
zbMATH Keywords: mixture model
zbMATH Keywords: rate of convergence
zbMATH Keywords: strong identifiability
zbMATH Keywords: pointwise rate
zbMATH Keywords: superefficiency
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6968601