Bayesian nonparametric spectral analysis of time series

Abstract

The past decades have seen increasing applications of nonparametric Bayesian technique to statistical inference problems. This is particularly phenomenal when it comes to the estimation of the spectral density function of a time series. However, most of the works regarding nonparametric Bayesian spectral analysis do not provide asymptotic theoretical justifications such as posterior consistency and posterior contraction rates. For those methods which do have theoretical justification, the Gaussianity assumption is often adopted. It is of interest to see whether such a method will still work if the Gaussianity assumption fails. In this thesis, we first prove two general theorems which allows us to establish asymptotic results in a rather convenient manner under various situations such as dependent data, infinite-dimensional parameter space and model misspecification. With the help of the general theorems, we are able to extend the proof of posterior consistency regarding the corrected Whittle likelihood proposed in the past literature to the case that the underlying time series is non-Gaussian. Moreover, a new approach called BDP-DW method which aims at estimating the time-varying spectral density function of a locally stationary time series is proposed with asymptotic properties obtained by utilizing the proposed general theorems. The BDP-DW method has been implemented in the beyondWhittle R package.