Adaptive nonparametric drift estimation for diffusion processes using Faber–Schauder expansions

F. van der Meulen; M. Schauer; J. van Waaij

doi:10.1007/s11203-017-9163-7

Adaptive nonparametric drift estimation for diffusion processes using Faber–Schauder expansions

F. van der Meulen
, M. Schauer
, J. van Waaij

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

We consider the problem of nonparametric estimation of the drift of a continuously observed one-dimensional diffusion with periodic drift. Motivated by computational considerations, van der Meulen et al. (Comput Stat Data Anal 71:615–632, 2014) defined a prior on the drift as a randomly truncated and randomly scaled Faber–Schauder series expansion with Gaussian coefficients. We study the behaviour of the posterior obtained from this prior
from a frequentist asymptotic point of view. If the true data generating drift is smooth, it is proved that the posterior is adaptive with posterior contraction rates for the L2-norm that are
optimal up to a log factor. Contraction rates in L p-norms with p ∈ (2,∞] are derived as well.

Original language	English
Pages (from-to)	603-628
Number of pages	26
Journal	Statistical Inference for Stochastic Processes
Volume	21
Issue number	3
Early online date	19 Jun 2017
DOIs	https://doi.org/10.1007/s11203-017-9163-7
Publication status	Published - 2018
Externally published	Yes

Funding

This work was partly supported by the Netherlands Organisation for Scientific Research (NWO) under the research programme ?Foundations of nonparametric Bayes procedures?, 639.033.110 and by the ERC Advanced Grant ?Bayesian Statistics in Infinite Dimensions?, 320637. Acknowledgements This work was partly supported by the Netherlands Organisation for Scientific Research (NWO) under the research programme \u201CFoundations of nonparametric Bayes procedures\u201D, 639.033.110 and by the ERC Advanced Grant \u201CBayesian Statistics in Infinite Dimensions\u201D, 320637.

Funders	Funder number
ERC advanced
NWO	639.033.110
European Commission	320637

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abstract = "We consider the problem of nonparametric estimation of the drift of a continuously observed one-dimensional diffusion with periodic drift. Motivated by computational considerations, van der Meulen et al. (Comput Stat Data Anal 71:615–632, 2014) defined a prior on the drift as a randomly truncated and randomly scaled Faber–Schauder series expansion with Gaussian coefficients. We study the behaviour of the posterior obtained from this prior from a frequentist asymptotic point of view. If the true data generating drift is smooth, it is proved that the posterior is adaptive with posterior contraction rates for the L2-norm that are optimal up to a log factor. Contraction rates in L p-norms with p ∈ (2,∞] are derived as well.",

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N2 - We consider the problem of nonparametric estimation of the drift of a continuously observed one-dimensional diffusion with periodic drift. Motivated by computational considerations, van der Meulen et al. (Comput Stat Data Anal 71:615–632, 2014) defined a prior on the drift as a randomly truncated and randomly scaled Faber–Schauder series expansion with Gaussian coefficients. We study the behaviour of the posterior obtained from this prior from a frequentist asymptotic point of view. If the true data generating drift is smooth, it is proved that the posterior is adaptive with posterior contraction rates for the L2-norm that are optimal up to a log factor. Contraction rates in L p-norms with p ∈ (2,∞] are derived as well.

AB - We consider the problem of nonparametric estimation of the drift of a continuously observed one-dimensional diffusion with periodic drift. Motivated by computational considerations, van der Meulen et al. (Comput Stat Data Anal 71:615–632, 2014) defined a prior on the drift as a randomly truncated and randomly scaled Faber–Schauder series expansion with Gaussian coefficients. We study the behaviour of the posterior obtained from this prior from a frequentist asymptotic point of view. If the true data generating drift is smooth, it is proved that the posterior is adaptive with posterior contraction rates for the L2-norm that are optimal up to a log factor. Contraction rates in L p-norms with p ∈ (2,∞] are derived as well.

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JO - Statistical Inference for Stochastic Processes

JF - Statistical Inference for Stochastic Processes

IS - 3

ER -

Adaptive nonparametric drift estimation for diffusion processes using Faber–Schauder expansions

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