Bayesian bootstrapping for symmetric distributions

Abstract

In this paper, we describe a Bayesian nonparametric approach to make inference for a spherically symmetric distribution. We consider a Dirichlet invariant process prior on the set of all spherically symmetric distributions and we derive the Dirichlet invariant process posterior. Indeed, our approach is an extension of the Dirichlet invariant process to a spherically symmetric distribution. In addition, we obtain the Dirichlet invariant process posterior for the infinite transformation group and we prove that it approaches the mixtures of Dirichlet processes. Moreover, we develop our approach to obtain the Bayesian nonparametric posterior distribution for functionals of the distribution's support when the support is symmetric with respect to the parallel lines of axes. This suggests a Bayesian nonparametric bootstrapping scheme. The estimates can be derived based on posterior averaging. Then, our simulation results demonstrate that our suggested bootstrapping technique improves the accuracy of the estimates.