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Mathematics > Statistics Theory

arXiv:1807.07237 (math)
[Submitted on 19 Jul 2018 (v1), last revised 13 Apr 2019 (this version, v2)]

Title:Optimal estimation of Gaussian mixtures via denoised method of moments

Authors:Yihong Wu, Pengkun Yang
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Abstract:The Method of Moments [Pea94] 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 non-uniqueness of statistically meaningful solutions, as well as the high computational cost of solving large polynomial systems. Moreover, theoretical analysis 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 ensures 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 [Han82] which involves non-convex optimization and lacks theoretical guarantees.
Subjects: Statistics Theory (math.ST)
Cite as: arXiv:1807.07237 [math.ST]
  (or arXiv:1807.07237v2 [math.ST] for this version)
  https://doi.org/10.48550/arXiv.1807.07237

Submission history

From: Pengkun Yang [view email]
[v1] Thu, 19 Jul 2018 04:47:21 UTC (156 KB)
[v2] Sat, 13 Apr 2019 19:36:54 UTC (164 KB)
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