Consistent non-parametric bayesian estimation for a time-inhomogeneous brownian motion
ESAIM: Probability and Statistics, Tome 18 (2014), pp. 332-341.

We establish posterior consistency for non-parametric Bayesian estimation of the dispersion coefficient of a time-inhomogeneous Brownian motion.

DOI : 10.1051/ps/2013039
Classification : 62G20, 62M05
Mots-clés : dispersion coefficient, non-parametric bayesian estimation, posterior consistency, time-inhomogeneous brownian motion 
@article{PS_2014__18__332_0,
     author = {Gugushvili, Shota and Spreij, Peter},
     title = {Consistent non-parametric bayesian estimation for a time-inhomogeneous brownian motion},
     journal = {ESAIM: Probability and Statistics},
     pages = {332--341},
     publisher = {EDP-Sciences},
     volume = {18},
     year = {2014},
     doi = {10.1051/ps/2013039},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2013039/}
}
TY  - JOUR
AU  - Gugushvili, Shota
AU  - Spreij, Peter
TI  - Consistent non-parametric bayesian estimation for a time-inhomogeneous brownian motion
JO  - ESAIM: Probability and Statistics
PY  - 2014
SP  - 332
EP  - 341
VL  - 18
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2013039/
DO  - 10.1051/ps/2013039
LA  - en
ID  - PS_2014__18__332_0
ER  - 
%0 Journal Article
%A Gugushvili, Shota
%A Spreij, Peter
%T Consistent non-parametric bayesian estimation for a time-inhomogeneous brownian motion
%J ESAIM: Probability and Statistics
%D 2014
%P 332-341
%V 18
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2013039/
%R 10.1051/ps/2013039
%G en
%F PS_2014__18__332_0
Gugushvili, Shota; Spreij, Peter. Consistent non-parametric bayesian estimation for a time-inhomogeneous brownian motion. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 332-341. doi : 10.1051/ps/2013039. http://archive.numdam.org/articles/10.1051/ps/2013039/

[1] A. Barron, M.J. Schervish and L. Wasserman, The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 (1999) 536-561. | MR | Zbl

[2] N. Choudhuri, S. Ghosal and A. Roy, Bayesian estimation of the spectral density of a time series. J. Amer. Statist. Assoc. 99 (2004) 1050-1059. | MR | Zbl

[3] P. Diaconis and D. Freedman, On the consistency of Bayes estimates. With a discussion and a rejoinder by the authors. Ann. Statist. 14 (1986) 1-67. | MR | Zbl

[4] V. Genon-Catalot and J. Jacod, On the estimation of the diffusion coefficient for multi-dimensional diffusion processes. Ann. Inst. Henri Poincaré Probab. Statist. 29 (1993) 119-151. | Numdam | MR | Zbl

[5] V. Genon-Catalot, C. Laredo and D. Picard, Nonparametric estimation of the diffusion coefficient by wavelets methods. Scand. J. Statist. 19 (1992) 317-335. | MR | Zbl

[6] S. Ghosal, J.K. Ghosh and R.V. Ramamoorthi, Consistency issues in Bayesian nonparametrics. Asymptotics, Nonparametrics, and Time Series. Vol. 158 of Textbooks Monogr. Dekker, New York (1999) 639-667. | MR | Zbl

[7] S. Ghosal and Y. Tang, Bayesian consistency for Markov processes. Sankhyā 68 (2006) 227-239. | MR | Zbl

[8] S. Gugushvili and P. Spreij, Non-parametric Bayesian drift estimation for stochastic differential equations (2012). Preprint arXiv:1206.4981 [math.ST]. | MR

[9] M. Hoffmann, Minimax estimation of the diffusion coefficient through irregular samplings. Statist. Probab. Lett. 32 (1997) 11-24. | MR | Zbl

[10] I.A. Ibragimov and R.Z. Has′minskiĭ, Asimptoticheskaya teoriya otsenivaniya [Asymptotic Theory of Estimation] (Russian). Nauka, Moscow (1979). | MR | Zbl

[11] F. Van Der Meulen, M. Schauer and H. Van Zanten, Reversible jump MCMC for nonparametric drift estimation for diffusion processes. Comput. Statist. Data Anal. 71 (2014) 615-632. Available on http://dx.doi.org/10.1016/j.csda.2013.03.002. | MR

[12] F.H. Van Der Meulen, A.W. Van Der Vaart and J.H. Van Zanten, Convergence rates of posterior distributions for Brownian semimartingale models. Bernoulli 12 (2006) 863-888. | MR | Zbl

[13] F. Van Der Meulen and H. Van Zanten, Consistent nonparametric Bayesian estimation for discretely observed scalar diffusions. Bernoulli 19 (2013) 44-63. | MR | Zbl

[14] L. Panzar and H. Van Zanten, Nonparametric Bayesian inference for ergodic diffusions. J. Statist. Plann. Inference 139 (2009) 4193-4199. | MR | Zbl

[15] O. Papaspiliopoulos, Y. Pokern, G.O. Roberts and A.M. Stuart, Nonparametric estimation of diffusions: a differential equations approach. Biometrika 99 (2012) 511-531. | MR

[16] G.A. Pavliotis, Y. Pokern and A.M. Stuart, Parameter estimation for multiscale diffusions: an overview. Statistical Methods for Stochastic Differential Equations. Vol. 124 of Monogr. Statist. Appl. Probab. CRC Press, Boca Raton, FL (2012) 429-472. | MR

[17] Y. Pokern, A.M. Stuart and J.H. Van Zanten. Posterior consistency via precision operators for nonparametric drift estimation in SDEs. Stoch. Process. Appl. 123 (2013) 603-628. | MR | Zbl

[18] L. Schwartz, On Bayes procedures. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965) 10-26. | MR | Zbl

[19] P. Soulier, Nonparametric estimation of the diffusion coefficient of a diffusion process. Stochastic Anal. Appl. 16 (1998) 185-200. | MR | Zbl

[20] A.W. Van Der Vaart, Asymptotic Statistics. Vol. 3 of Cambr. Ser. Stat. Probab. Math. Cambridge University Press, Cambridge (1998). | MR | Zbl

[21] A.W. Van Der Vaart and J.H. Van Zanten, Rates of contraction of posterior distributions based on Gaussian process priors. Ann. Statist. 36 (2008a) 1435-1463. | MR | Zbl

[22] A.W. Van Der Vaart and J.H. Van Zanten, Reproducing kernel Hilbert spaces of Gaussian priors. Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh. Vol. 3 of Inst. Math. Stat. Collect. Inst. Math. Statist., Beachwood, OH (2008) 200-222. | MR

[23] S. Walker, On sufficient conditions for Bayesian consistency. Biometrika 90 (2003) 482-488. | MR | Zbl

[24] S. Walker, New approaches to Bayesian consistency. Ann. Statist. 32 (2004) 2028-2043. | MR | Zbl

[25] L. Wasserman, Asymptotic properties of nonparametric Bayesian procedures. Practical Nonparametric and Semiparametric Bayesian Statistics. Vol. 133 of Lect. Notes Statist. Springer, New York (1998) 293-304. | MR | Zbl

[26] H. Van Zanten, Nonparametric Bayesian methods for one-dimensional diffusion models. Math. Biosci. (2013). Available on http://dx.doi.org/10.1016/j.mbs.2013.03.008. | MR | Zbl

Cité par Sources :