Diffusions with measurement errors. I. Local asymptotic normality

Gloter, Arnaud; Jacod, Jean

Gloter, Arnaud ; Jacod, Jean

ESAIM: Probability and Statistics, Tome 5 (2001), pp. 225-242.

Résumé

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 $1 / \sqrt{n}$ (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 $n^{- 1 / 4}$ .

MR Zbl | 4 citations dans Numdam

Classification : 60J60, 62F12, 62M05
Mots clés : statistics of diffusions, measurement errors, LAN property

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     author = {Gloter, Arnaud and Jacod, Jean},
     title = {Diffusions with measurement errors. {I.} {Local} asymptotic normality},
     journal = {ESAIM: Probability and Statistics},
     pages = {225--242},
     publisher = {EDP-Sciences},
     volume = {5},
     year = {2001},
     mrnumber = {1875672},
     zbl = {1008.60089},
     language = {en},
     url = {http://www.numdam.org/item/PS_2001__5__225_0/}
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Gloter, Arnaud; Jacod, Jean. Diffusions with measurement errors. I. Local asymptotic normality. ESAIM: Probability and Statistics, Tome 5 (2001), pp. 225-242. http://www.numdam.org/item/PS_2001__5__225_0/

Bibliographie
Cité par

[1] P. Bickel and Y. Ritov, Inference in hidden Markov models. I. Local asymptotic normality in the stationary case. Bernoulli 2 (1996) 199-228. | MR | Zbl

[2] P. Bickel and Y. Ritov, Asymptotic normality for the maximum likelihood estimator for general hidden Markov models. Ann. Statist. 26 (1998) 1614-1635. | MR | Zbl

[3] G. Dohnal, On estimating the diffusion coefficient. J. Appl. Probab. 24 (1987) 105-114. | MR | Zbl

[4] V. Genon-Catalot and J. Jacod, On the estimation of the diffusion coefficient for multidimensional diffusion processes. Ann. Inst. H. Poincaré Probab. Statist. 29 (1993) 119-153. | Numdam | Zbl

[5] V. Genon-Catalot and J. Jacod, Estimation of the diffusion coefficient for diffusion processes: random sampling. Scand. J. Statist. 21 (1994) 193-221. | Zbl

[6] A. Gloter and J. Jacod, Diffusion with measurement error. II. Optimal estimators (2000). | Numdam | MR | Zbl

[7] E. Gobet, LAMN property for elliptic diffusions (2000).

[8] J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin (1987). | MR | Zbl

[9] J.L. Jensen and N. Petersen, Asymptotic normality of the Maximum likelihood estimator in state space models. Ann. Statist. 27 (1999) 514-535. | MR | Zbl

[10] B. Leroux, Maximum likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 (1992) 127-143. | MR | Zbl

[11] L. Lecam and G.L. Yang, Asymptotics in Statistics. Springer-Verlag, Berlin (1990). | MR | Zbl

[12] M.B. Malyutov and O. Bayborodin, Fitting diffusion and trend in noise via Mercer expansion, in Proc. 7th Int. Conf. on Analytical and Stochastic Modeling Techniques. Hamburg (2000).

[13] T. Ryden, Consistent and asymptotically normal estimators for hidden Markov models. Ann. Statist. 22 (1994) 1884-1895. | MR | Zbl