Diffusions with measurement errors. I. Local Asymptotic Normality

E-ISSN： 1262-3318|5|issue|225-242

ISSN： 1292-8100

Source： ESAIM: Probability and Statistics, Vol.5, Iss.issue, 2010-03, pp. : 225-242

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Abstract

We consider a diffusion process X which is observed at times i/nfor i = 0,1,...,n, each observation being subject to a measurementerror. All errors are independent and centered Gaussian with knownvariance p_n. There is an unknown parameter within the diffusioncoefficient, to be estimated. In this first paper thecase when X is indeed a Gaussian martingale is examined: we can provethat the LAN property holds under quite weak smoothness assumptions,with an explicit limiting Fisher information. What is perhaps the mostinteresting is the rate at which this convergence takes place:it is $1/\sqrt{n}$ (as when there is no measurement error) when p_n goes fastenough to 0, namely np_n is bounded. Otherwise, and provided thesequence p_n itself is bounded, the rate is (p_n / n)^1/4. Inparticular if p_n = p does not depend on n, we get a rate n^-1/4.

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