# Diffusions with measurement errors. I. Local asymptotic normality

ESAIM: Probability and Statistics (2001)

- Volume: 5, page 225-242
- ISSN: 1292-8100

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topGloter, Arnaud, and Jacod, Jean. "Diffusions with measurement errors. I. Local asymptotic normality." ESAIM: Probability and Statistics 5 (2001): 225-242. <http://eudml.org/doc/104275>.

@article{Gloter2001,

abstract = {We consider a diffusion process $X$ which is observed at times $i/n$ for $i=0,1,\ldots ,n$, each observation being subject to a measurement error. All errors are independent and centered gaussian with known variance $\rho _n$. There is an unknown parameter within the diffusion coefficient, to be estimated. In this first paper the case when $X$ is indeed a gaussian martingale is examined: we can prove that the LAN property holds under quite weak smoothness assumptions, with an explicit limiting Fisher information. What is perhaps the most interesting is the rate at which this convergence takes place: it is $1/\sqrt\{n\}$ (as when there is no measurement error) when $\rho _n$ goes fast enough to $0$, namely $n\rho _n$ is bounded. Otherwise, and provided the sequence $\rho _n$ itself is bounded, the rate is $(\rho _n/n)^\{1/4\}$. In particular if $\rho _n=\rho $ does not depend on $n$, we get a rate $n^\{-1/4\}$.},

author = {Gloter, Arnaud, Jacod, Jean},

journal = {ESAIM: Probability and Statistics},

keywords = {statistics of diffusions; measurement errors; LAN property},

language = {eng},

pages = {225-242},

publisher = {EDP-Sciences},

title = {Diffusions with measurement errors. I. Local asymptotic normality},

url = {http://eudml.org/doc/104275},

volume = {5},

year = {2001},

}

TY - JOUR

AU - Gloter, Arnaud

AU - Jacod, Jean

TI - Diffusions with measurement errors. I. Local asymptotic normality

JO - ESAIM: Probability and Statistics

PY - 2001

PB - EDP-Sciences

VL - 5

SP - 225

EP - 242

AB - We consider a diffusion process $X$ which is observed at times $i/n$ for $i=0,1,\ldots ,n$, each observation being subject to a measurement error. All errors are independent and centered gaussian with known variance $\rho _n$. There is an unknown parameter within the diffusion coefficient, to be estimated. In this first paper the case when $X$ is indeed a gaussian martingale is examined: we can prove that the LAN property holds under quite weak smoothness assumptions, with an explicit limiting Fisher information. What is perhaps the most interesting is the rate at which this convergence takes place: it is $1/\sqrt{n}$ (as when there is no measurement error) when $\rho _n$ goes fast enough to $0$, namely $n\rho _n$ is bounded. Otherwise, and provided the sequence $\rho _n$ itself is bounded, the rate is $(\rho _n/n)^{1/4}$. In particular if $\rho _n=\rho $ does not depend on $n$, we get a rate $n^{-1/4}$.

LA - eng

KW - statistics of diffusions; measurement errors; LAN property

UR - http://eudml.org/doc/104275

ER -

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## Citations in EuDML Documents

top- M. Hoffmann, A. Munk, J. Schmidt-Hieber, Adaptive wavelet estimation of the diffusion coefficient under additive error measurements
- Sophie Donnet, Adeline Samson, Parametric inference for mixed models defined by stochastic differential equations
- Arnaud Gloter, Emmanuel Gobet, LAMN property for hidden processes : the case of integrated diffusions

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