Diffusions with measurement errors. I. Local asymptotic normality

Arnaud Gloter; Jean Jacod

ESAIM: Probability and Statistics (2001)

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

Abstract

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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 ρ 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 / n (as when there is no measurement error) when ρ n goes fast enough to 0 , namely n ρ n is bounded. Otherwise, and provided the sequence ρ n itself is bounded, the rate is ( ρ n / n ) 1 / 4 . In particular if ρ n = ρ does not depend on n , we get a rate n - 1 / 4 .

How to cite

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Gloter, 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 -

References

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