The Bernstein-von Mises Theorem for Stationary Processes

Tamaki, Kenichiro

doi:10.14490/jjss.38.311

Cited by 5 publications

(5 citation statements)

References 9 publications

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“…Remark The practice of using a (relatively) compact set as the parameter space in (frequentist/Bayesian/parametric/ nonparametric) inference of spectral density functions has a long history (e.g. Hannan, 1973; Taniguchi, 1981; Tamaki, 2008; Bardet et al, 2008; Meier et al, 2020). The reason we adopt this practice is that the proof of the uniform LLNs (Assumption 3) on an infinite‐dimensional space will benefit substantially from the application of some Arzelà‐Ascoli type theorems (e.g.…”

Section: Bayesian Modeling and Posterior Consistencymentioning

confidence: 99%

“…The practice of using a (relatively) compact set as the parameter space in (frequentist/Bayesian/parametric/ nonparametric) inference of spectral density functions has a long history (e.g. Hannan, 1973;Taniguchi, 1981;Tamaki, 2008;Bardet et al, 2008;Meier et al, 2020).…”

Section: Parameter Spacementioning

confidence: 99%

“…An ensuing question is: Will the posterior distribution associated with a pseudo‐likelihood still be consistent in the sense that the posterior distribution eventually concentrates in the neighborhood of the true parameter, namely the underlying spectral density function in this case, as the sample size gets large? For parametric Bayesian modeling using the Whittle likelihood, the Bernstein‐von Mises theorem (Tamaki, 2008) guarantees that under some regularity conditions, Bayesian and frequentist inference procedures can produce the same asymptotically correct result. In the case of nonparametric Bayesian modeling, however, posterior consistency needs to be proved explicitly.…”

Section: Introductionmentioning

confidence: 99%

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Posterior consistency for the spectral density of non‐Gaussian stationary time series

Tang

Kirch

Lee

et al. 2023

Scandinavian J Statistics

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Various nonparametric approaches for Bayesian spectral density estimation of stationary time series have been suggested in the literature, mostly based on the Whittle likelihood approximation. A generalization of this approximation involving a nonparametric correction of a parametric likelihood has been proposed in the literature with a proof of posterior consistency for spectral density estimation in combination with the Bernstein–Dirichlet process prior for Gaussian time series. In this article, we will extend the posterior consistency result to non‐Gaussian time series by employing a general consistency theorem for dependent data and misspecified models. As a special case, posterior consistency for the spectral density under the Whittle likelihood is also extended to non‐Gaussian time series. Small sample properties of this approach are illustrated with several examples of non‐Gaussian time series.

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Section: Bayesian Modeling and Posterior Consistencymentioning

confidence: 99%

Section: Parameter Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Posterior consistency for the spectral density of non‐Gaussian stationary time series

Tang

Kirch

Lee

et al. 2023

Scandinavian J Statistics

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“…Moreover, the asymptotic behaviour of the Bayesian estimators are also guaranteed by the ARMA(1, 1) property of the process (y i ). If the prior density function is continuous and positive in an open neighbourhood of the real parameters, the Bayesian estimators are asymptotically normal (see [32] in which a generalized Bernstein Von Mises theorem for stationary "short memory" processes is given, or [27] for a discussion on the Bayesian analysis of ARMA processes).…”

Section: Performance Of Statistical Estimatorsmentioning

confidence: 99%

Forecasting Trends with Asset Prices

2015

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Abstract. In this paper, we consider a stochastic asset price model where the trend is an unobservable Ornstein Uhlenbeck process. We first review some classical results from Kalman filtering. Expectedly, the choice of the parameters is crucial to put it into practice. For this purpose, we obtain the likelihood in closed form, and provide two on-line computations of this function. Then, we investigate the asymptotic behaviour of statistical estimators. Finally, we quantify the effect of a bad calibration with the continuous time mis-specified Kalman filter. Numerical examples illustrate the difficulty of trend forecasting in financial time series. Motivations

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“…cases. Lemma 1 is also based on a series of Lipschitz assumptions20 concerning the prior π and the Fisher information matrix I(θ ). These assumptions are satisfied for classical distributions21 (e.g.…”

mentioning

confidence: 99%

Weak convergence of posteriors conditional on maximum pseudo-likelihood estimates and implications in ABC

Soubeyrand

Haon-Lasportes

2015

Statistics & Probability Letters

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The Bernstein-von Mises Theorem for Stationary Processes

Abstract: This paper discusses the asymptotic properties of the posterior density under Whittle measure. The Bernstein-von Mises theorem is shown for short-and longmemory stationary processes. Applications to Bayesian inference for time series are provided.

Cited by 5 publications

References 9 publications

Posterior consistency for the spectral density of non‐Gaussian stationary time series

Posterior consistency for the spectral density of non‐Gaussian stationary time series

Forecasting Trends with Asset Prices

Weak convergence of posteriors conditional on maximum pseudo-likelihood estimates and implications in ABC

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