Bayesian semi-parametric estimation of the long-memory parameter under FEXP-priors

In this paper we study the semi-parametric problem of the estimation of the long-memory parameter d in a Gaussian long-memory model. Considering a family of priors based on FEXP models, called FEXP priors in Rousseau et al. (2012), we derive concentration rates together with a Bernstein-von Mises theorem for the posterior distribution of d, under Sobolev regularity conditions on the short-memory part of the spectral density. Three different variations on the FEXP priors are studied. We prove that one of them leads to the minimax (up to a logn term) posterior concentration rate for d, under Sobolev conditions on the short memory part of the spectral density, while the other two lead to sub-optimal posterior concentration rates in d. Interestingly these results are contrary to those obtained in Rousseau et al. (2012) for the global estimation of the spectral density.