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 upper and lower bounds for the contraction rates of the 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 Bayes posterior distributions.