Bayesian nonparametric estimation of the spectral density of a long or intermediate memory Gaussian process

TL;DR

This paper introduces a Bayesian nonparametric method for estimating the spectral density of Gaussian processes with long or intermediate memory, demonstrating posterior consistency and convergence rates without using Whittle's approximation.

Contribution

It presents a novel Bayesian approach for spectral density estimation that ensures posterior consistency and convergence rates, applicable to long-range dependent Gaussian processes.

Findings

Proves posterior consistency for both $d$ and $g$

Establishes convergence rates for a broad class of priors

Applies the method to fractionally exponential priors

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.