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December 2018 Strong identifiability and optimal minimax rates for finite mixture estimation
Philippe Heinrich, Jonas Kahn
Ann. Statist. 46(6A): 2844-2870 (December 2018). DOI: 10.1214/17-AOS1641

Abstract

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 Chen [Ann. Statist. 23 (1995) 221–233].

By contrast, it turns out that there are estimators with a (nonuniform) pointwise rate of estimation of $n^{-1/2}$ for all mixing distributions with a finite number of components.

Citation

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Philippe Heinrich. Jonas Kahn. "Strong identifiability and optimal minimax rates for finite mixture estimation." Ann. Statist. 46 (6A) 2844 - 2870, December 2018. https://doi.org/10.1214/17-AOS1641

Information

Received: 1 July 2015; Revised: 1 October 2017; Published: December 2018
First available in Project Euclid: 7 September 2018

zbMATH: 06968601
MathSciNet: MR3851757
Digital Object Identifier: 10.1214/17-AOS1641

Subjects:
Primary: 62G05
Secondary: 62G20

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 6A • December 2018
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