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August 2020 Optimal estimation of Gaussian mixtures via denoised method of moments
Yihong Wu, Pengkun Yang
Ann. Statist. 48(4): 1981-2007 (August 2020). DOI: 10.1214/19-AOS1873

Abstract

The method of moments (Philos. Trans. R. Soc. Lond. Ser. A 185 (1894) 71–110) is one of the most widely used methods in statistics for parameter estimation, by means of solving the system of equations that match the population and estimated moments. However, in practice and especially for the important case of mixture models, one frequently needs to contend with the difficulties of non-existence or nonuniqueness of statistically meaningful solutions, as well as the high computational cost of solving large polynomial systems. Moreover, theoretical analyses of the method of moments are mainly confined to asymptotic normality style of results established under strong assumptions.

This paper considers estimating a $k$-component Gaussian location mixture with a common (possibly unknown) variance parameter. To overcome the aforementioned theoretic and algorithmic hurdles, a crucial step is to denoise the moment estimates by projecting to the truncated moment space (via semidefinite programming) before solving the method of moments equations. Not only does this regularization ensure existence and uniqueness of solutions, it also yields fast solvers by means of Gauss quadrature. Furthermore, by proving new moment comparison theorems in the Wasserstein distance via polynomial interpolation and majorization techniques, we establish the statistical guarantees and adaptive optimality of the proposed procedure, as well as oracle inequality in misspecified models. These results can also be viewed as provable algorithms for generalized method of moments (Econometrica 50 (1982) 1029–1054) which involves nonconvex optimization and lacks theoretical guarantees.

Citation

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Yihong Wu. Pengkun Yang. "Optimal estimation of Gaussian mixtures via denoised method of moments." Ann. Statist. 48 (4) 1981 - 2007, August 2020. https://doi.org/10.1214/19-AOS1873

Information

Received: 1 July 2018; Revised: 1 April 2019; Published: August 2020
First available in Project Euclid: 14 August 2020

MathSciNet: MR4134783
Digital Object Identifier: 10.1214/19-AOS1873

Subjects:
Primary: 62G05
Secondary: 62C20

Keywords: Deconvolution , Finite mixture model , Gauss quadrature , Gaussian mixture , method of moments , Minimax optimality , moment space , semidefinite programming , Wasserstein distance

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 4 • August 2020
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