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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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.

Article information

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

Dates
First available in Project Euclid: 18 July 2012

Permanent link to this document
http://projecteuclid.org/euclid.aos/1342625458

Digital Object Identifier
doi:10.1214/11-AOS955

Mathematical Reviews number (MathSciNet)
MR2985940

Zentralblatt MATH identifier
1274.62340

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

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

Citation

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. http://projecteuclid.org/euclid.aos/1342625458.


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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.