Bayesian nonparametric estimation of the spectral density of a long memory Gaussian time series
Abstract
Let $\mathbf {X}=\{X_t, t=1,2,... \}$ be a stationary Gaussian random process, with mean $EX_t=\mu$ and covariance function $\gamma(\tau)=E(X_t\mu)(X_{t+\tau}\mu)$. Let $f(\lambda)$ be the corresponding spectral density; a stationary Gaussian process is said to be longrange dependent, if the spectral density $f(\lambda)$ can be written as the product of a slowly varying function $\tilde{f}(\lambda)$ and the quantity $\lambda ^{2d}$. In this paper we propose a novel Bayesian nonparametric approach to the estimation of the spectral density of $\mathbf {X}$. We prove that, under some specific assumptions on the prior distribution, our approach assures posterior consistency both when $f(\cdot)$ and $d$ are the objects of interest. The rate of convergence of the posterior sequence depends in a significant way on the structure of the prior; we provide some general results and also consider the fractionally exponential (FEXP) family of priors (see below). Since it has not a well founded justification in the long memory setup, we avoid using the Whittle approximation to the likelihood function and prefer to use the true Gaussian likelihood.
 Publication:

arXiv eprints
 Pub Date:
 November 2007
 arXiv:
 arXiv:0711.0876
 Bibcode:
 2007arXiv0711.0876R
 Keywords:

 Mathematics  Statistics;
 62G20 (Primary);
 62M15 (Secondary)
 EPrint:
 Submitted to the Electronic Journal of Statistics (http://www.ijournals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org)