Optimal empirical Bayes estimation for the Poisson model via minimum-distance methods

The Robbins estimator is the most iconic and widely used procedure in the empirical Bayes literature for the Poisson model. On the one hand, this method has been recently shown to be minimax optimal in terms of the regret (excess risk over the Bayesian oracle that knows the true prior) for various non-parametric classes of priors. On the other hand, it has been long recognized in practice that the Robbins estimator lacks the desired smoothness and monotonicity of Bayes estimators and can be easily derailed by those data points that were rarely observed before. Based on the minimum-distance distance method, we propose a suite of empirical Bayes estimators, including the classical non-parametric maximum likelihood, that outperform the Robbins method in a variety of synthetic and real data sets and retain its optimality in terms of minimax regret.