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Bayesian nonparametric spectral density estimation using B-spline priors. (English) Zbl 1430.62072

Summary: We present a new Bayesian nonparametric approach to estimating the spectral density of a stationary time series. A nonparametric prior based on a mixture of B-spline distributions is specified and can be regarded as a generalization of the Bernstein polynomial prior of S. Petrone [Scand. J. Stat. 26, No. 3, 373–393 (1999; Zbl 0939.62046); Can. J. Stat. 27, No. 1, 105–126 (1999; Zbl 0929.62044)] and N. Choudhuri et al. [J. Am. Stat. Assoc. 99, No. 468, 1050–1059 (2004; Zbl 1055.62100)]. Whittle’s likelihood approximation is used to obtain the pseudo-posterior distribution. This method allows for a data-driven choice of the number of mixture components and the location of knots. Posterior samples are obtained using a Metropolis-within-Gibbs Markov chain Monte Carlo algorithm, and mixing is improved using parallel tempering. We conduct a simulation study to demonstrate that for complicated spectral densities, the B-spline prior provides more accurate Monte Carlo estimates in terms of \(L_1\)-error and uniform coverage probabilities than the Bernstein polynomial prior. We apply the algorithm to annual mean sunspot data to estimate the solar cycle. Finally, we demonstrate the algorithm’s ability to estimate a spectral density with sharp features, using real gravitational wave detector data from LIGO’s sixth science run, recoloured to match the Advanced LIGO target sensitivity.

MSC:

62G07 Density estimation
62M15 Inference from stochastic processes and spectral analysis
85A25 Radiative transfer in astronomy and astrophysics
65D07 Numerical computation using splines
83C35 Gravitational waves
62-08 Computational methods for problems pertaining to statistics
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