Geodesic
Bernstein-von Mises Theorem and small noise asymptotics of Bayes estimators for parabolic stochastic partial differential equations
Teoriâ slučajnyh processov, Tome 23 (2018) no. 1, pp. 6-17.

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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) driven by space-time white noise as the intensity of noise decreases to zero. As a consequence, the Bayes estimators of the drift parameter, for smooth loss functions and priors, are shown to be strongly consistent and asymptotically normal, asymptotically efficient and asymptotically equivalent to the maximum likelihood estimator as the intensity of noise decreases to zero. Also computable pseudo-posterior density and pseudo-Bayes estimators based on finite dimensional projections are shown to have similar asymptotics as the noise decreases to zero and the dimension of the projection remains fixed.
Keywords: stochastic partial differential equations, cylindrical Brownian motion, Bernstein-von Mises theorem, Bayes estimator, consistency, asymptotic normality, small noise. 
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Jaya P. N. Bishwal. Bernstein-von Mises Theorem and small noise asymptotics of Bayes estimators for parabolic stochastic partial differential equations. Teoriâ slučajnyh processov, Tome 23 (2018) no. 1, pp. 6-17. http://geodesic.mathdoc.fr/item/THSP_2018_23_1_a1/

[1] J. P. N. Bishwal, “Bayes and sequential estimation in Hilbert space valued stochastic differential equations”, J. Korean Statistical Society, 28:1 (1999), 93–106 | MR

[2] J. P. N. Bishwal, “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 | DOI | MR | Zbl

[3] J. P. N. Bishwal, “The Bernstein-von Mises theorem and spectral asymptotics of Bayes estimators for parabolic SPDEs”, J. Australian Math. Society, 72:2 (2002), 287–298 | DOI | MR | Zbl

[4] J. P. N. Bishwal, “A new estimating function for discretely sampled diffusions”, Random Operators and Stochastic Equations, 15:1 (2007), 65–88 | DOI | MR | Zbl

[5] J. P. N. Bishwal, Parameter Estimation in Stochastic Differential Equations, Lecture Notes in Mathematics, 1923, Springer-Verlag, 2008 | DOI | MR | Zbl

[6] J. D. Borwanker, G. Kallianpur, B. L. S. Prakasa Rao, “The Bernstein-von Mises theorem for Markov processes”, Ann. Math. Statist., 42 (1971), 1241–1253 | DOI | MR | Zbl

[7] A. Bose, “The Bernstein-von Mises theorem for a certain class of diffusion processes”, Sankhyā Ser. A, 45 (1983), 150–160 | MR

[8] M. Huebner, “A characterization of asymptotic behaviour of maximum likelihood estimators for stochastic PDE's”, Math. Methods. Statist., 6 (1997), 395–415 | MR | Zbl

[9] M. Huebner, “Asymptotic properties of the maximum likelihood estimator for stochastic PDEs disturbed by small noise”, Statistical Inference for Stochastic Processes, 2 (1999), 57–68 | DOI | MR | Zbl

[10] M. Huebner, R. Z. Khasminskii, B. L. Rozovskii, “Two examples of parameter estimation for stochastic partial differential equations”, Stochastic Processes, Freschrift in Honour of G. Kallianpur, eds. S. Cambanis, J. K. Ghosh, R. L. Karandikar, P. K. Sen, Springer, Berlin, 1992, 149–160 | MR

[11] M. Huebner, B. L. Rozovskii, “On the asymptotic properties of maximum likelihood estimators for parabolic stochastic PDEs”, Prob. Theor. Rel. Fields, 103 (1995), 143–163 | DOI | MR | Zbl

[12] I. A. Ibragimov, R. Z. Has'minskii, Statistical Estimation: Asymptotic Theory, Springer-Verlag, Berlin, 1981 | MR | Zbl

[13] I. A. Ibragimov, R. Z. Khasminskii, “Estimation problems for coefficients of stochastic partial differential equations, Part I”, Theory Probab. Appl., 43 (1998), 370–387 | DOI | MR | Zbl

[14] T. Koski, W. Loges, “Asymptotic statistical inference for a stochastic heat flow problem”, Statist. Prob. Letters, 3 (1985), 185–189 | DOI | MR | Zbl

[15] T. Koski, W. Loges, “On minimum contrast estimation for Hilbert space valued stochastic differential equations”, Stochastics, 16 (1986), 217–225 | DOI | MR | Zbl

[16] Le Cam, G. L. Yang, Asymptotics in Statistics : Some Basic Concepts, Springer, New York, 2000 | MR | Zbl

[17] W. Loges, “Girsanov's theorem in Hilbert space and an application to the statistics of Hilbert space-valued stochastic differential equations”, Stoch. Proc. Appl., 17 (1984), 243–263 | DOI | MR | Zbl

[18] S. V. Lototsky, B. L. Rozovskii, “Spectral asymptotics of some functionals arising in statistical inference for SPDEs”, Stoch. Process. Appl., 79 (1999), 69–94 | DOI | MR | Zbl