## The Annals of Statistics

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

#### 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

https://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

#### 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. https://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.