Open Access
December 2019 Beyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis
Claudia Kirch, Matthew C. Edwards, Alexander Meier, Renate Meyer
Bayesian Anal. 14(4): 1037-1073 (December 2019). DOI: 10.1214/18-BA1126


Nonparametric Bayesian inference has seen a rapid growth over the last decade but only few nonparametric Bayesian approaches to time series analysis have been developed. Most existing approaches use Whittle’s likelihood for Bayesian modelling of the spectral density as the main nonparametric characteristic of stationary time series. It is known that the loss of efficiency using Whittle’s likelihood can be substantial. On the other hand, parametric methods are more powerful than nonparametric methods if the observed time series is close to the considered model class but fail if the model is misspecified. Therefore, we suggest a nonparametric correction of a parametric likelihood that takes advantage of the efficiency of parametric models while mitigating sensitivities through a nonparametric amendment. We use a nonparametric Bernstein polynomial prior on the spectral density with weights induced by a Dirichlet process and prove posterior consistency for Gaussian stationary time series. Bayesian posterior computations are implemented via an MH-within-Gibbs sampler and the performance of the nonparametrically corrected likelihood for Gaussian time series is illustrated in a simulation study and in three astronomy applications, including estimating the spectral density of gravitational wave data from the Advanced Laser Interferometer Gravitational-wave Observatory (LIGO).

Version Information

A previous version of the Supplementary material contained proofs of Lemma 1.2 and Theorem 2 that relied on an incorrect deduction. The current version amends the proofs. In addition, an additional assumption for the validity of Lemma 1.2 has been included to require that Sn are symmetric positive definite matrices with uniformly bounded eigenvalues μi(Sn), i.e. there exist constants m and M such that 0<mμi(Sn)M< for i=1,,n and all n. The statement of Theorem 2 in the main text remains unchanged.


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Claudia Kirch. Matthew C. Edwards. Alexander Meier. Renate Meyer. "Beyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis." Bayesian Anal. 14 (4) 1037 - 1073, December 2019.


Published: December 2019
First available in Project Euclid: 30 October 2018

zbMATH: 1435.62332
MathSciNet: MR4044845
Digital Object Identifier: 10.1214/18-BA1126

Keywords: Bayesian nonparametrics , Frequency domain , laser interferometric gravitational wave data , Spectral density , stationary time series

Vol.14 • No. 4 • December 2019
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