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

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

2010-07-22

arxiv.org/abs/1007.3823v1

Statistics

Methodology

Scientific paper

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. We propose a novel Bayesian nonparametric approach for the estimation of the spectral density of such processes. Within this approach, 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 function, and does not resort to Whittle's approximation, which is not valid in a long memory set-up.

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