Estimation of the Mixing Distribution for a Normal Mean with Applications to the Compound Decision Problem

Edelman, David

doi:10.1214/aos/1176351056

Cited by 21 publications

(11 citation statements)

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“…In contrast, denoised method of moments is efficiently computable and adaptively achieves the optimal rate of accuracy as given in Theorem 2. For arbitrary Gaussian location mixtures in one dimension, the minimum distance estimator was considered in [Ede88] in the context of empirical Bayes. Under the assumptions of bounded first moment, it is shown in [Ede88, Corollary 2] that the mixing distribution can be estimated at rate O((log n) −1/4 ) under the L 2 -distance between the CDFs; this loss is, however, weaker than the W 1 -distance (i.e.…”

Section: Related Workmentioning

confidence: 99%

Optimal estimation of Gaussian mixtures via denoised method of moments

Wu¹,

Yang²

2018

Preprint

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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 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 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.

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Section: Related Workmentioning

confidence: 99%

Optimal estimation of Gaussian mixtures via denoised method of moments

Wu¹,

Yang²

2018

Preprint

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“…Also, results for the compound decision problem are extended to the nonGaussian case by exploiting Berry-Esseen type results for the regression coefficients; this leads to our minimax results. Finally, permutation arguments are used to extend an insight of Edelman (1988) in the Gaussian compound decision problem to show that the empirical Bayes estimator is also minimum risk equivariant.…”

Section: Introductionmentioning

confidence: 99%

Empirical Bayes Forecasts of One Time Series Using Many Predictors

Knox¹,

Stock²,

Watson³

2001

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“…When the x i 's are not observed, but only their (noisy) measurements y i are available, then Π(x) is to be estimated from y 1 , ...y h . The reader may consult also Edelman (1988) where further discussion concerning the estimation of Π as well as theoretical results on the resulting estimator's properties are derived.…”

Section: Compound Decision and Estimation Problems And Exchangeabilitymentioning

confidence: 99%

Regularization and Bayesian learning in dynamical systems: Past, present and future

Chiuso

2016

Annual Reviews in Control

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Regularization and Bayesian methods for system identification have been repopularized in the recent years, and proved to be competitive w.r.t. classical parametric approaches. In this paper we shall make an attempt to illustrate how the use of regularization in system identification has evolved over the years, starting from the early contributions both in the Automatic Control as well as Econometrics and Statistics literature. In particular we shall discuss some fundamental issues such as compound estimation problems and exchangeability which play and important role in regularization and Bayesian approaches, as also illustrated in early publications in Statistics. The historical and foundational issues will be given more emphasis (and space), at the expense of the more recent developments which are only briefly discussed. The main reason for such a choice is that, while the recent literature is readily available, and surveys have already been published on the subject, in the author's opinion a clear link with past work had not been completely clarified.

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Estimation of the Mixing Distribution for a Normal Mean with Applications to the Compound Decision Problem

Cited by 21 publications

References 0 publications

Optimal estimation of Gaussian mixtures via denoised method of moments

Optimal estimation of Gaussian mixtures via denoised method of moments

Empirical Bayes Forecasts of One Time Series Using Many Predictors

Regularization and Bayesian learning in dynamical systems: Past, present and future

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