The Annals of Statistics
- Ann. Statist.
- Volume 45, Number 2 (2017), 833-865.
Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator
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.
Ann. Statist. Volume 45, Number 2 (2017), 833-865.
Received: April 2015
Revised: March 2016
First available in Project Euclid: 16 May 2017
Permanent link to this document
Digital Object Identifier
Zentralblatt MATH identifier
Primary: 62G20: Asymptotic properties 62G05: Estimation 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 62G08: Nonparametric regression 62G07: Density estimation
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
- 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.