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The Bernstein-von Mises theorem and spectral asymptotics of Bayes estimators for parabolic SPDEs

Part of: Inference from stochastic processes Parametric inference

Published online by Cambridge University Press:  09 April 2009

J. P. N. Bishwal
J. P. N. Bishwal
Affiliation:
Department of Economics, Fisher Hall, Princeton University, Princeton, NJ 08544-1021, USA e-mail: [email protected]
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Abstract

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The Bernstein-von Mises theorem, concerning the convergence of suitably normalized and centred posterior density to normal density, is proved for a certain class of linearly parametrized parabolic stochastic partial differential equations (SPDEs) as the number of Fourier coefficients in the expansion of the solution increases to infinity. As a consequence, the Bayes estimators of the drift parameter, for smooth loss functions and priors, are shown to be strongly consistent, asymptotically normal and locally asymptotically minimax (in the Hajek-Le Cam sense), and asymptotically equivalent to the maximum likelihood estimator as the number of Fourier coefficients become large. Unlike in the classical finite dimensional SDEs, here the total observation time and the intensity of noise remain fixed.

Keywords

stochastic partial differential equationsdiffusion fieldBernstein-von Mises theoremBayes estimatorconsistencyasymptotic normalitylocal asymptotic minimaxityasymptotic equivalencespectral theory.

MSC classification

Secondary: 62M05: Markov processes 62F12: Asymptotic properties of estimators 62F15: Bayesian inference
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 2 , April 2002 , pp. 287 - 298
Copyright
Copyright © Australian Mathematical Society 2002

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