## The Annals of Statistics

### Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator

#### Abstract

We consider the asymptotic behaviour of the marginal maximum likelihood empirical Bayes posterior distribution in general setting. First, we characterize the set where the maximum marginal likelihood estimator is located with high probability. Then we provide oracle type of upper and lower bounds for the contraction rates of the empirical Bayes posterior. We also show that the hierarchical Bayes posterior achieves the same contraction rate as the maximum marginal likelihood empirical Bayes posterior. We demonstrate the applicability of our general results for various models and prior distributions by deriving upper and lower bounds for the contraction rates of the corresponding empirical and hierarchical Bayes posterior distributions.

#### Article information

Source
Ann. Statist., Volume 45, Number 2 (2017), 833-865.

Dates
Revised: March 2016
First available in Project Euclid: 16 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.aos/1494921959

Digital Object Identifier
doi:10.1214/16-AOS1469

Mathematical Reviews number (MathSciNet)
MR3650402

Zentralblatt MATH identifier
1371.62048

#### Citation

Rousseau, Judith; Szabo, Botond. Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator. Ann. Statist. 45 (2017), no. 2, 833--865. doi:10.1214/16-AOS1469. https://projecteuclid.org/euclid.aos/1494921959

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#### Supplemental materials

• Supplement to “Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator”. This is the supplementary material associated to the present paper. We provide here the proofs of Propositions 3.1–3.6, together with some technical Lemmas used in the context of priors (T2) and (T3) and some technical Lemmas used in the study of the hierarchical Bayes posteriors. Finally some Lemmas used in the regression and density estimation problems are given.