The Annals of Statistics

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

Judith Rousseau, Nicolas Chopin, and Brunero Liseo

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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.

Article information

Ann. Statist., Volume 40, Number 2 (2012), 964-995.

First available in Project Euclid: 18 July 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties
Secondary: 62M15: Spectral analysis

Bayesian nonparametric consistency FEXP priors Gaussian long memory processes rates of convergence


Rousseau, Judith; Chopin, Nicolas; Liseo, Brunero. Bayesian nonparametric estimation of the spectral density of a long or intermediate memory Gaussian process. Ann. Statist. 40 (2012), no. 2, 964--995. doi:10.1214/11-AOS955.

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Supplemental materials

  • Supplementary material: Bayesian nonparametric estimation of the spectral density of a long or intermediate memory Gaussian process: Supplementary material. Proof of technical lemmas and theorems stated in the paper.