# Diffusions with measurement errors. I. Local Asymptotic Normality

ESAIM: Probability and Statistics (2010)

- 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 (2010): 225-242. <http://eudml.org/doc/116585>.

@article{Gloter2010,

abstract = {
We consider a diffusion process X which is observed at times i/n
for i = 0,1,...,n, each observation being subject to a measurement
error. All errors are independent and centered Gaussian with known
variance pn. 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 pn goes fast
enough to 0, namely npn is bounded. Otherwise, and provided the
sequence pn itself is bounded, the rate is (pn / n)1/4. In
particular if pn = p 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},

month = {3},

pages = {225-242},

publisher = {EDP Sciences},

title = {Diffusions with measurement errors. I. Local Asymptotic Normality},

url = {http://eudml.org/doc/116585},

volume = {5},

year = {2010},

}

TY - JOUR

AU - Gloter, Arnaud

AU - Jacod, Jean

TI - Diffusions with measurement errors. I. Local Asymptotic Normality

JO - ESAIM: Probability and Statistics

DA - 2010/3//

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,...,n, each observation being subject to a measurement
error. All errors are independent and centered Gaussian with known
variance pn. 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 pn goes fast
enough to 0, namely npn is bounded. Otherwise, and provided the
sequence pn itself is bounded, the rate is (pn / n)1/4. In
particular if pn = p 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/116585

ER -

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