Rates of convergence for Bayes estimators of mixtures of exponential power densities

We consider estimating densities that are location or location-scale mixtures of kernels in the family of exponential power distributions, which includes the Laplace and normal distributions. We focus on rates of convergence for the Hellinger distance of Bayes’ estimators. The problem has been recently investigated for normal mixtures under the severe restriction that the scale parameter stays bounded away from zero and infinity. We consider location and location-scale mixtures of exponential power densities, without any assumption on the scale parameter. We show that the Bayes’ estimator corresponding to a Dirichlet mixture of exponential power distributions converges at nearly parametric rate except for a logarithmic term, the power of the log-term depending on the tail behaviour of the priors for the location and scale parameters.