Choudhuri, Nidhan; Ghosal, Subhashis; Roy, Anindya Bayesian estimation of the spectral density of a time series. (English) Zbl 1055.62100 J. Am. Stat. Assoc. 99, No. 468, 1050-1059 (2004). Summary: 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 [see P. W. Whittle, Bull. Inst. Int. Stat. 39, 105–129 (1962; Zbl 0116.11403)]. 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 L. Schwartz [Z. Wahrscheinlichkeitstheor. Verw. Geb. 4, 10–26 (1965; Zbl 0158.17606)] 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. Cited in 2 ReviewsCited in 52 Documents MSC: 62M15 Inference from stochastic processes and spectral analysis 62F15 Bayesian inference 65C40 Numerical analysis or methods applied to Markov chains Keywords:Bernstein polynomial; Dirichlet process; Metropolis algorithm; periodogram; posterior consistency; posterior distribution Citations:Zbl 0116.11403; Zbl 0158.17606 PDFBibTeX XMLCite \textit{N. Choudhuri} et al., J. Am. Stat. Assoc. 99, No. 468, 1050--1059 (2004; Zbl 1055.62100) Full Text: DOI