Bayesian nonparametric estimation of the spectral density of a long or intermediate memory Gaussian process
Judith Rousseau, Nicolas Chopin, Brunero Liseo
The Annals of Statistics · 2012 · Volume 40 · Issue 2
Abstract
A stationary Gaussian process is said to be long-range dependent (resp., anti-persistent) if its spectral density $f(\lambda)$ can be written as $f(\lambda)=|\lambda|^{-2d}g(|\lambda|)$, where $0<d<1/2$ (resp., $-1/2<d<0$), and $g$ is continuous and positive. We propose a novel Bayesian nonparametric approach for the estimation of the spectral density of such processes. We prove posterior consistency for both $d$ and $g$, under appropriate conditions on the prior distribution. We establish the rate of convergence for a general class of priors and apply our results to the family of fractionally exponential priors. Our approach is based on the true likelihood and does not resort to Whittle’s approximation.
Keywords
62G20, 62M15, Bayesian nonparametric, consistency, FEXP priors, Gaussian long memory processes, rates of convergence
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