Bayesian Estimation of the Spectral Density of a Time Series

Abstract

This article describes a Bayesian approach to estimating the spectral density of a stationary time series. A nonparametric prior on the spectral density is described through Bernstein polynomials. Because the actual likelihood is very complicated, a pseudoposterior distribution is obtained by updating the prior using the Whittle likelihood. A Markov chain Monte Carlo algorithm for sampling from this posterior distribution is described that is used for computing the posterior mean, variance, and other statistics. A consistency result is established for this pseudoposterior distribution that holds for a short-memory Gaussian time series and under some conditions on the prior. To prove this asymptotic result, a general consistency theorem of Schwartz is extended for a triangular array of independent, nonidentically distributed observations. This extension is also of independent interest. A simulation study is conducted to compare the proposed method with some existing methods. The method is illustrated with the well-studied sunspot dataset.

Bayesian Methods for Function Estimation

“…Thus Schwartz's theorem is the right tool for studying consistency in semiparametric models. Extensions of Schwartz's consistency theorem to independent, nonidentically distributed observations have been obtained by Amewou-Atisso et al (2003) and Choudhuri et al (2004a). The former does not use sieves and hence is useful only when weak topology is put on the infinite-dimensional part of the parameter.…”

“…Using the contiguity result of Choudhuri et al (2004c), the following result was shown by Choudhuri et al (2004a) under the above assumptions.…”

“…Consistency in probability instead of that in the almost sure sense allows certain relaxations in the condition to be verified. Choudhuri et al (2004a) considered such a formulation which is described below. …”

“…Normalizing f to q = f/τ with the normalizing constant τ = f , Choudhuri et al (2004a) induced a prior on f by first putting a Bernstein polynomial prior on q and then putting an independent prior on τ . Thus, the prior on f is described by the following hierarchical scheme:…”

On some aspects of the asymptotic properties of Bayesian approaches in nonparametric and semiparametric models

“…The study of asymptotic properties of Bayesian nonparametric methods methods was initiated by the seminal papers of [Schwartz, 1965,Barron, 1988 then increased significantly after the works of [Barron et al, 1999, Ghosal et al, 2000a. Since then posterior concentration has been extensively studied in various types of models including nonparametric regression [Ghosal and van der Vaart, 2007], Markov models [Tang and Ghosal, 2007], Gaussian time series [Choudhuri et al, 2004. In this paper, we present some recent advances in the study of frequentist properties of Bayesian nonparametric inference.…”