We consider the problem of non-parametric estimation of the deterministic dispersion coefficient of a linear stochastic differential equation based on discrete time observations on its solution. We take a Bayesian approach to the problem and under suitable regularity assumptions derive the posteror contraction rate. This rate turns out to be the optimal posterior contraction rate.
Accepté le :
DOI : 10.1051/ps/2016008
Mots-clés : Dispersion coefficient, non-parametric Bayesian estimation, posterior contraction rate, stochastic differential equation
@article{PS_2016__20__143_0,
author = {Gugushvili, Shota and Spreij, Peter},
title = {Posterior contraction rate for non-parametric {Bayesian} estimation of the dispersion coefficient of a stochastic differential equation},
journal = {ESAIM: Probability and Statistics},
pages = {143--153},
publisher = {EDP-Sciences},
volume = {20},
year = {2016},
doi = {10.1051/ps/2016008},
mrnumber = {3528621},
zbl = {1357.62200},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2016008/}
}
TY - JOUR AU - Gugushvili, Shota AU - Spreij, Peter TI - Posterior contraction rate for non-parametric Bayesian estimation of the dispersion coefficient of a stochastic differential equation JO - ESAIM: Probability and Statistics PY - 2016 SP - 143 EP - 153 VL - 20 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2016008/ DO - 10.1051/ps/2016008 LA - en ID - PS_2016__20__143_0 ER -
%0 Journal Article %A Gugushvili, Shota %A Spreij, Peter %T Posterior contraction rate for non-parametric Bayesian estimation of the dispersion coefficient of a stochastic differential equation %J ESAIM: Probability and Statistics %D 2016 %P 143-153 %V 20 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2016008/ %R 10.1051/ps/2016008 %G en %F PS_2016__20__143_0
Gugushvili, Shota; Spreij, Peter. Posterior contraction rate for non-parametric Bayesian estimation of the dispersion coefficient of a stochastic differential equation. ESAIM: Probability and Statistics, Tome 20 (2016), pp. 143-153. doi : 10.1051/ps/2016008. http://archive.numdam.org/articles/10.1051/ps/2016008/
S.A. van de Geer, Applications of Empirical Process Theory. Vol. 6 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2000). | MR | Zbl
, and , Nonparametric estimation of the diffusion coefficient by wavelets methods. Scand. J. Statist. 19 (1992) 317–335. | MR | Zbl
, and , Convergence rates of posterior distributions. Ann. Statist. 28 (2000) 500–531. | DOI | MR | Zbl
and , Convergence rates of posterior distributions for non-i.i.d. observations. Ann. Statist. 35 (2007) 192–223. | DOI | MR | Zbl
S. Ghosal, J.K. Ghosh and R.V. Ramamoorthi, Non-informative priors via sieves and packing numbers. Advances in Statistical Decision Theory and Applications, Stat. Ind. Technol. Birkhäuser Boston, Boston, MA (1997) 119–132. | MR | Zbl
S. Ghosal, J.K. Ghosh and R.V. Ramamoorthi, Consistency issues in Bayesian nonparametrics. Asymptotics, Nonparametrics, and Time Series. Vol. 158 of Statist. Textbooks Monogr. Dekker, New York (1999) 639–667. | MR | Zbl
and , Non-parametric Bayesian estimation of a dispersion coefficient of the stochastic differential equation. ESAIM: PS 18 (2014) 332–341. | Zbl
, Minimax estimation of the diffusion coefficient through irregular samplings. Statist. Probab. Lett. 32 (1997) 11–24. | DOI | MR | Zbl
I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Vol. 113 of Graduate Texts in Mathematics. Springer-Verlag, New York (1988). | MR | Zbl
and , Misspecification in infinite-dimensional Bayesian statistics, Ann. Statist. 34 (2006) 837–877. | DOI | MR | Zbl
R. Nickl and J. Söhl, Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions. Preprint arXiv:1510.05526 [math.ST] (2015). | MR
C.E. Rasmussen and C.K.I. Williams, Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. MIT Press, Cambridge, MA (2006). | MR | Zbl
and , Rates of convergence of posterior distributions. Ann. Statist. 29 (2001) 687–714. | DOI | MR | Zbl
, Nonparametric estimation of the diffusion coefficient of a diffusion process. Stochastic Anal. Appl. 16 (1998) 185–200. | DOI | MR | Zbl
A.B. Tsybakov, Introduction to Nonparametric Estimation. Springer Series in Statistics. Revised and extended from the 2004 French original. Translated by Vladimir Zaiats. Springer, New York (2009). | MR | Zbl
and , Rates of contraction of posterior distributions based on Gaussian process priors. Ann. Statist. 36 (2008) 1435–1463. | DOI | MR | Zbl
J. van Waaij and H. van Zanten, Gaussian process methods for one-dimensional diffusions: optimal rates and adaptation. Preprint arXiv:1506.00515 [math.ST] (2015). | MR
L. Wasserman, Asymptotic properties of nonparametric Bayesian procedures. In Practical Nonparametric and Semiparametric Bayesian Statistics. Vol. 133 of Lect. Notes Stat. Springer, New York (1998) 293–304. | MR | Zbl
and , Probability inequalities for likelihood ratios and convergence rates of sieve MLEs. Ann. Statist. 23 (1995) 339–362. | MR | Zbl
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