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Abstract
Consider the Gaussian sequence model under the additional assumption that a fixed fraction of the means is known. We study the problem of variance estimation from a frequentist Bayesian perspective. The maximum likelihood estimator (MLE) for σ^{2} is biased and inconsistent. This raises the question whether the posterior is able to correct the MLE in this case. By developing a new proving strategy that uses refined properties of the posterior distribution, we find that the marginal posterior is inconsistent for any i.i.d. prior on the mean parameters. In particular, no assumption on the decay of the prior needs to be imposed. Surprisingly, we also find that consistency can be retained for a hierarchical prior based on Gaussian mixtures. In this case we also establish a limiting shape result and determine the limit distribution. In contrast to the classical Bernsteinvon Mises theorem, the limit is nonGaussian. We show that the Bayesian analysis leads to new statistical estimators outperforming the correctly calibrated MLE in a numerical simulation study.
Original language  English 

Pages (fromto)  239271 
Number of pages  33 
Journal  Electronic Journal of Statistics 
Volume  14 
Issue number  1 
DOIs  
Publication status  Published  8 Jan 2020 
Keywords
 Bernsteinvon Mises theorems
 Frequentist Bayes
 Gaussian sequence model
 Maximum likelihood
 Semiparametric inference
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Bayesian variance estimation in the Gaussian sequence model withpartial information on the means
Gianluca Finocchio (Speaker)
21 Dec 2019Activity: Talk or presentation › Invited talk