Posterior rates of convergence for Dirichlet mixtures of exponential power densities

Abstract

A Dirichlet mixture of exponential power distributions, as a prior on densities supported on the real line in the problem of Bayesian density estimation, is a natural generalization of a Dirichlet mixture of normals, which has been shown to possess good frequentist asymptotic properties in terms of posterior consistency and rates of convergence. In this article, we establish upper bounds on the rates of convergence for the posterior distribution of a Dirichlet mixture of exponential power densities, assuming that the true density has the same form as the model. When the kernel is analytic and the mixing distribution has either compact support or sub-exponential tails, a nearly parametric rate, up to a logarithmic factor whose exponent depends on the tail behaviour of the base measure of the Dirichlet process and the exponential decay rate at zero of the prior for the scale parameter, is obtained. The result covers the important special case where the true density is a location mixture of normals and shows that a nearly parametric rate arises also when the prior on the scale contains zero in its support, provided it has a sufficiently fast decay rate at zero. This improves on some recent results on density estimation with Dirichlet mixtures of normals by allowing the inverse-gamma distribution, which is a commonly used prior on the square of the bandwidth. When the kernel is not infinitely differentiable at zero, as the case may be depending on the shape parameter, the posterior distribution is shown to concentrate around the sampling density at a slower rate.