General maximum likelihood empirical Bayes estimation of normal means

By Wenhua Jiang, Cun-Hui Zhang. 2009

252 citations recorded by the source corpus.

We propose a general maximum likelihood empirical Bayes (GMLEB) method for the estimation of a mean vector based on observations with i.i.d. normal errors. We prove that under mild moment conditions on the unknown means, the average mean squared error (MSE) of the GMLEB is within an infinitesimal fraction of the minimum average MSE among all separable estimators which use a single deterministic estimating function on individual observations, provided that the risk is of greater order than (log n) 5 /n. We also prove that the GMLEB is uniformly approximately minimax in regular and weak l p balls when the order of the length-normalized norm of the unknown means is between (log n) k1 /n 1/(p^2) and n/(log n) k2 . Simulation experiments demonstrate that the GMLEB outperforms the James―Stein and several state-of-the-art threshold estimators in a wide range of settings without much down side.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/faba7e21a29fa8f438e8c12bebf6c7bd49b94b3a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Statistics and Probability

Citations in the corpus

The Poisson Compound Decision Problem Revisited (2010, methodology): “arametric maximum-likelihood (NPMLE), as suggested by Kiefer and Wolfowitz (1956), is an alternative approach for estimating δG. It yields, automatically, a monotone and smooth decision function. See Jiang and Zhang (2009) for the normal model. To simplify the discussion, we will assume that λ1,...,”
Uncertainty Quantification for the Horseshoe (with Discussion) (2016, background): “Despite the ubiquity of problems with sparse structures, and the large amount of research effort into finding consistent and minimax optimal estimators for the underlying sparse structures [44, 22, 13, 12, 20, 18, 21, 15, 8, 5, 4, 34], the number of options for uncertainty quantification in the spar”
NONPARAMETRIC EMPIRICAL BAYES AND COMPOUND DECISION APPROACHES TO ESTIMATION OF A HIGH-DIMENSIONAL VECTOR OF NORMAL MEANS (2009, methodology): “An approach involving MLE estimation of G was recently suggested by Wenhua and Zhang (2007).”
Group-Linear Empirical Bayes Estimates for a Heteroscedastic Normal Mean (2015, methodology): “Then, as presented, for example, by Jiang and Zhang (2009), for a separable estimator with θ̂i = t(Xi), Rn(θ, θ̂) = n∑ i=1 1 n Eθi[t(Xi) − θi]2 = E[t(Y ) − ξ ]2 (4) where the expectation in the last term is taken over the pair (Y, ξ ) of random variables jointly distributed according to ξ ∼ 1 n n∑…”
Adaptive posterior contraction rates for the horseshoe (2017, background): “Those most directly related to this work are [30, 20, 9, 8, 18, 16, 19, 14, 5, 3, 2, 25].”
The Horseshoe Estimator: Posterior Concentration around Nearly Black Vectors (2014, background): “…the minimax rate in ℓ q norm, q ∈ (0 , 2], for mean vectors that are nearly black, and in ℓ q norm, q ∈ (0 , 2) for mean vectors of bounded weak ℓ p norm, p ∈ (0 , q ). includes Other work on empirical Bayes approaches to the two-group model (Efron, 2008; Jiang and Zhang, 2009; Yuan and Lin, 2005).”
Asymptotically minimax empirical Bayes estimation of a sparse normal mean vector (2013, methodology): “…to estimate the mean vector θ = ( θ 1 , . . . , θ n ) under squared-error loss k ˆ θ − θ k 2 , where k · k is the usual ℓ 2 -norm on R n (e.g., Abramovich et al. 2006; Brown and Greenshtein 2009; Castillo and van der Vaart 2012; Donoho and Johnstone 1994; Donoho et al. 1992; Jiang and Zhang 2009).”
False discovery rates: a new deal (2016, unknown): “In particular, [35,37] consider the unconstrained non-parametric maximum likelihood estimate (NPMLE) for g.”