Bayesian nonparametric estimation of the spectral density of a long memory Gaussian time series

Let X = {Xt, t = 1, 2, . . . } be a stationary Gaussian random process, with mean EXt = and covariance function γ(τ ) = E(Xt − )(Xt+τ − ). Let f(λ) be the corresponding spectral density; a stationary Gaussian process is said to be long-range dependent, if the spectral density f(λ) can be written as the product of a slowly varying function ˜ f(λ) and the quantity λ−2d. In this paper we propose a novel Bayesian nonparametric approach to the estimation of the spectral density of X. We prove that,under some specific assumptions on the prior distribution, our approach assures posterior consistency both when f(.) and d are the objects of interest. The rate of convergence of the posterior sequence depends in a significant way on the structure of the prior; we provide some general results and also consider the fractionally exponential (FEXP) family of priors (see below). Since it has not a well founded justification in the long memory set-up, we avoid using the Whittle approximation to the likelihood function and prefer to use the true Gaussian likelihood. It makes the computational burden of the method quite challenging. To mitigate the impact of that in finite sample computations, we propose to use a Population MonteCarlo (PMC) algorithm, which avoids rejecting some proposed values, as it regularly happens with MCMC algorithms. We also propose an extension of PMC in order to deal with the case of varying dimension parameter space. We finally present an application of our approach.ou