• Bernoulli
  • Volume 12, Number 5 (2006), 863-888.

Convergence rates of posterior distributions for Brownian semimartingale models

F.H. Van Der Meulen, Aad W. Van Der Vaart, and J.H. Van Zanten

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We consider the asymptotic behaviour of posterior distributions based on continuous observations from a Brownian semimartingale model. We present a general result that bounds the posterior rate of convergence in terms of the complexity of the model and the amount of prior mass given to balls centred around the true parameter. This result is illustrated for three special cases of the model: the Gaussian white-noise model, the perturbed dynamical system and the ergodic diffusion model. Some examples for specific priors are discussed as well.

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Bernoulli, Volume 12, Number 5 (2006), 863-888.

First available in Project Euclid: 23 October 2006

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Bayesian estimation continuous semimartingale Dirichlet process Hellinger distance infinite-dimensional model rate of convergence wavelets


Van Der Meulen, F.H.; Van Der Vaart, Aad W.; Van Zanten, J.H. Convergence rates of posterior distributions for Brownian semimartingale models. Bernoulli 12 (2006), no. 5, 863--888. doi:10.3150/bj/1161614950.

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  • [1] Birgé, L. and Massart, P. (2000) An adaptive compression algorithm in Besov spaces. Constr. Approx., 16, 1-36.
  • [2] Devore, R.A. and Lorentz, G.G. (1993) Constructive Approximation. Berlin: Springer-Verlag.
  • [3] Dietz, H.M. and Kutoyants, Yu.A. (2003) Parameter estimation for some non-recurrent solutions of SDE. Statist. Decisions, 21(1), 29-45.
  • [4] Ghosal, S. and van der Vaart, A.W. (2003) Posterior convergence rates of Dirichlet mixtures of normal distributions for smooth densities. Preprint 2003-12. publications.php (accessed 24 February 2006).
  • [5] Ghosal, S. and van der Vaart, A.W. (2004) Convergence rates of posterior distributions for noniid observations. Preprint 2003-17. (accessed 24 February 2006). Ann. Statist. To appear.
  • [6] Ghosh J.K. and Ramamoorthi, R.V. (2003) Bayesian Nonparametrics. New York: Springer-Verlag.
  • [7] Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998) Wavelets, Approximation, and Statistical Applications. Lectures Notes in Statist. 129. New York: Springer-Verlag.
  • [8] Höpfner, R. and Kutoyants, Yu.A. (2003) On a problem of statistical inference in null recurrent diffusions. Statist. Inference Stochastic Process., 6(1), 25-42.
  • [9] Ibragimov, I.A. and Has´minskii R.Z. (1981) Statistical Estimation: Asymptotic Theory. New York: Springer-Verlag.
  • [10] Itô, K. and McKean, H.P., Jr. (1965) Diffusion Processes and Their Sample Paths. Berlin: Springer- Verlag.
  • [11] Karatzas, I. and Shreve, S.E. (1991) Brownian Motion and Stochastic Calculus, 2nd edn. New York: Springer-Verlag.
  • [12] Kerkyacharian, G. and Picard, D, (2004) Entropy, universal coding, approximation and bases properties. Constr. Approx., 20, 1-37.
  • [13] Kutoyants, Yu.A. (1984) Parameter Estimation for Stochastic Processes. Berlin: Heldermann Verlag.
  • [14] Kutoyants, Yu.A. (1994) Identification of Dynamical Systems with Small Noise. Dordrecht: Kluwer Academic.
  • [15] Kutoyants, Yu.A. (2004) Statistical Inference for Ergodic Diffusion Processes. New York: Springer- Verlag.
  • [16] Liptser, R.S. and Shiryayev, A.N. (1977) Statistics of Random Processes I. New York: Springer-Verlag.
  • [17] Loukianova, D. and Loukianov, O. (2005) Deterministic equivalents of additive functionals of recurrent diffusions and drift estimation. Preprint.
  • [18] Pollard, D. (1990) Empirical Processes: Theory and Applications, NSF-CBMS Reg. Conf. Ser. Probab. Statist. 2. Hayward, CA: Institute of Mathematical Statistics; Alexandria, VA: American Statistical Association.
  • [19] Protter, P. (2004) Stochastic Integration and Differential Equations, 2nd edn. Berlin: Springer-Verlag.
  • [20] Prakasa Rao, B.L.S. (1999) Statistical Inference for Diffusion Type Processes. London: Edward Arnold.
  • [21] Revuz, D. and Yor, M. (1999) Continuous Martingales and Brownian Motion, 3rd edn. Berlin: Springer-Verlag.
  • [22] Shen, X. and Wasserman, L. (2001) Rates of convergence of posterior distributions. Ann. Statist., 29, 687-714.
  • [23] van der Vaart, A.W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes with Applications to Statistics. New York: Springer-Verlag.
  • [24] van Zanten, J.H. (2003) On uniform laws of large numbers for ergodic diffusions and consistency of estimators. Statist. Inference Stochastic Process., 6(2), 199-213.
  • [25] van Zanten, J.H. (2005) On the rate of convergence of the maximum likelihood estimator in Brownian semimartingale models. Bernoulli, 11, 643-664.
  • [26] Zhao, L.H. (2000) Bayesian aspects of some nonparametric problems. Ann. Statist., 28, 532-552.