Posterior consistency for the spectral density of non-Gaussian stationary time series

TL;DR

This paper extends Bayesian spectral density estimation methods to non-Gaussian stationary time series, proving posterior consistency and demonstrating small sample performance through examples.

Contribution

It generalizes existing posterior consistency results from Gaussian to non-Gaussian time series using Shalizi's theorem, including the Whittle likelihood approach.

Findings

Posterior consistency is established for non-Gaussian time series.

The approach is validated with small sample examples.

Extension of the Whittle likelihood to non-Gaussian data.

Abstract

Various nonparametric approaches for Bayesian spectral density estimation of stationary time series have been suggested in the literature, mostly based on the Whittle likelihood approximation. A generalization of this approximation has been proposed in Kirch et al. who prove posterior consistency for spectral density estimation in combination with the Bernstein-Dirichlet process prior for Gaussian time series. In this paper, we will extend the posterior consistency result to non-Gaussian time series by employing a general consistency theorem of Shalizi for dependent data and misspecified models. As a special case, posterior consistency for the spectral density under the Whittle likelihood as proposed by Choudhuri, Ghosal and Roy is also extended to non-Gaussian time series. Small sample properties of this approach are illustrated with several examples of non-Gaussian time series.