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

Part of: Parametric inference Inference from stochastic processes

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: jbishwal@princeton.edu
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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

References

[1]Basawa, I. V. and Rao, B. L. S. Prakasa, Statistical inference for stochastic processes (Academic Press, New York, 1980).Google Scholar
[2]Bishwal, J. P. N., ‘Bayes and sequential estimation in Hilbert space valued stochastic differential equations’, J. Korean Statist. Society 28 (1999), 93–106.Google Scholar
[3]Bishwal, J. P. N., ‘Rates of convergence of the posterior distributions and the Bayes estimators in the Ornstein-Uhlenbeck process’, Random Operators and Stochastic Equations 8 (2000), 51–70.CrossRefGoogle Scholar
[4]Bishwal, J. P. N., Markussen, B. and Sørensen, M., ‘Large deviations and Berry-Esseen inequality for the maximum likelihood estimator in parabolic SPDEs’, preprint, 2000.Google Scholar
[5]Bishwal, J. P. N. and Sørensen, M., ‘Sequential estimation in parabolic SPDEs’, preprint, 2000.Google Scholar
[6]Borwanker, J. D., Kallianpur, G. and Rao, B. L. S. Prakasa, ‘The Bernstein-von Mises theorem for Markov processes’, Ann. Math. Statist. 42 (1971), 1241–1253.CrossRefGoogle Scholar
[7]Bose, A., ‘The Bernstein-von Mises theorem for a certain class of diffusion processes’, Sankhyā Ser.A 45 (1983), 150–160.Google Scholar
[8]Cam, L. Le, ‘On some asymptotic properties of maximum likelihood and related Bayes estimators’, Univ. Calif., Publ. in Statist. 1 (1953), 227–330.Google Scholar
[9]Cam, L. Le and Yang, G. L., Asymptotics in statistics: some basic concepts (Springer, New York, 1990).CrossRefGoogle Scholar
[10]Carmona, R. and Rozovskii, B. L., Stochastic partial differential equations: six perspectives (Amer. Math. Soc., Rhode Island, 1999).CrossRefGoogle Scholar
[11]Ghosal, S., Ghosh, J. K. and Samanta, T., ‘On convergence of posterior distributions’, Ann. Statist. 23 (1995), 2145–2152.CrossRefGoogle Scholar
[12]Holden, H., Øksendal, B., Ubøe, J. and Zhang, T., Stochastic partial differential equations (Birkhäuser, Boston, 1996).CrossRefGoogle Scholar
[13]Huebner, M., ‘A characterization of asymptotic behaviour of maximum likelihood estimators for stochastic PDEs’, Math. Methods Statist. 6 (1997), 395–415.Google Scholar
[14]Huebner, M., Has'minskii, R. Z. and Rozovskii, B. L., ‘Two examples of parameter estimation for stochastic partial differential equations’, in: Stochastic processes (eds. Cambanis, S., Ghosh, J. K., Karandikar, R. L. and Sen, P. K.), Festschrift in Honour of G. Kallianpur (Springer, Berlin, 1993) pp. 149–160.CrossRefGoogle Scholar
[15]Huebner, M. and Rozovskii, B. L., ‘On the asymptotic properties of maximum likelihood estimators for parabolic stochastic PDEs’, Probab. Theory Related Fields 103 (1995), 143–163.CrossRefGoogle Scholar
[16]Ibragimov, I. A. and Khasminskii, R. Z., Statistical estimation: asymptotic theory (Springer, Berlin, 1981).CrossRefGoogle Scholar
[17]Ibragimov, I. A. and Khasminskii, B. Z., ‘Estimation problems for coefficients of stochastic partial differential equations. Part I’, Theory Probab. Appl. 43 (1998), 370–387.CrossRefGoogle Scholar
[18]Itô, K., Foundations of stochastic differential equations in infinite dimensional spaces, CBMS-NSF Regional Conference Series in Appl. Math. 49 (SIAM, Philadelphia, Pennsylvania, 1984).CrossRefGoogle Scholar
[19]Kallianpur, G. and Xiong, X., Infinite dimensional stochastic differential equations, Lecture Notes-Monograph Series 26 (Institute of Mathematical Statistics, Hayward, California, 1995).CrossRefGoogle Scholar
[20]Koski, T. and Loges, W., ‘Asymptotic statistical inference for a stochastic heat flow problem’, Statist. Prob. Letters 3 (1985), 185–189.CrossRefGoogle Scholar
[21]Koski, T. and Loges, W., ‘On minimum contrast estimation for Hilbert space valued stochastic differential equations’, Stochastics 16 (1986), 217–225.CrossRefGoogle Scholar
[22]Kutoyants, Yu. A., Parameter estimation for stochastic processes, (Translated and edited by Rao, B. L. S. Prakasa) (Heldermann, Berlin, 1984).Google Scholar
[23]Kutoyants, Yu. A., Identification of dynamical systems with small noise (Kluwer, Dordrecht, 1994).CrossRefGoogle Scholar
[24]Kutoyants, Yu. A., Parameter estimation for ergodic diffusion processes, unpublished monograph, 1999.Google Scholar
[25]Liptser, R. S. and Shiryayev, A. N., Statistics of random processes I, II (Springer, Berlin, 1977, 1978).Google Scholar
[26]Loges, W., ‘Girsanov's theorem in Hilbert space and an application to the statistics of Hilbert space valued stochastic differential equations’, Stochastic Process Appl. 17 (1984), 243–263.CrossRefGoogle Scholar
[27]Lototsky, S. V. and Rozovskii, B. L., ‘Spectral asymptotics of some functionals arising in statistical inference for SPDEs’, Stochastic Process Appl. 79 (1999), 69–94.CrossRefGoogle Scholar
[28]Mishra, M. N., ‘The Bernstein-von Mises theorem for a class of non-homogeneous diffusion processes’, Statist. Decisions 7 (1989), 153–165.Google Scholar
[29]Piterbarg, L. and Rozovskii, B. L., ‘Maximum likelihood estimators in the equations of physical oceanography’, in: Stochastic modelling in physical oceanography (eds. Adler, R., Muller, P. and Rozovskii, B.) (Birkhäuser, Boston, 1996) pp. 397–421.CrossRefGoogle Scholar
[30]Piterbarg, L. and Rozovskii, B. L., ‘On asymptotic problems of parameter estimation in stochastic PDEs: discrete time sampling’, Math. Methods Statist. 6 (1996), 200–223.Google Scholar
[31]Rao, B. L. S. Prakasa, ‘The Bernstein-von Mises theorem for a class of diffusion processes’, Theory Random Processes 9 (1980), 95–101 (in Russian).Google Scholar
[32]Rao, B. L. S. Prakasa, ‘On Bayes estimation for diffusion fields’, in: Statistics: applications and new directions, Proc. ISI Golden Jubilee Conferences (eds. Ghosh, J. K. and Roy, J.) (Statist. Publ. Soc., Calcutta, 1984) pp. 504–511.Google Scholar
[33]Rao, B. L. S. Prakasa, Statistical inference for diffusion type processes (Arnold, London, 1999).Google Scholar
[34]Walsh, J. B., An introduction to stochastic partial differential equations, Lecture Notes in Math. 1180 (Springer, Berlin, 1986) pp. 265–439.Google Scholar