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Bayesian estimation of the spectral density of a time series. (English) Zbl 1055.62100

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.

MSC:

62M15 Inference from stochastic processes and spectral analysis
62F15 Bayesian inference
65C40 Numerical analysis or methods applied to Markov chains
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