LAMN property for hidden processes : the case of integrated diffusions

Gloter, Arnaud; Gobet, Emmanuel

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 1, pp. 104-128.

Dans ce papier nous démontrons la propriété LAMN pour le modèle statistique constitué par l’observation des moyennes locales d’une diffusion $X$. Nos données sont définies comme ${\int }_0^1{X}_{(s+i)/n}{d}_{\mu }\left(s\right)$ avec $i=0,...,n-1$ et le paramètre inconnu apparaît seulement dans le coefficient de diffusion du processus $X$. Bien que cette observation ne soit ni gaussienne ni markovienne nous pouvons, par le calcul de Malliavin, obtenir une expression pour la log-vraisemblance du modèle. Nous sommes alors capables de calculer l’information asymptotique et montrons qu’elle est la même que pour l’observation ponctuelle de la diffusion.

In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process $X$. Our data are given by ${\int }_0^1{X}_{(s+i)/n}{d}_{\mu }\left(s\right)$ for $i=0,...,n-1$ and the unknown parameter appears in the diffusion coefficient of the process $X$ only. Although the data are neither markovian nor gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic expansion. We actually find that the asymptotic information of this model is the same one as for a usual discrete sampling of $X$.

O. E. Barndorff-Nielsen and N. Shephard. Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 167-241. | MR | Zbl

P. D. Ditlevsen, S. Ditlevsen and K. K. Andersen. The fast climate fluctuations during the stadial and interstadial climate states. Ann. Glaciology 35 (2002) 457-462.

S. Ditlevsen and M. Sørensen. Inference for observations of integrated diffusion processes. Scand. J. Statist. 31 (2004) 417-429. | MR | Zbl

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

V. Genon-Catalot and J. Jacod. On the estimation of the diffusion coefficient for multi-dimensional processes. Ann. Inst. H. Poincaré Probab. Statist. 29 (1993) 119-151. | Numdam | MR | Zbl

V. Genon-Catalot, T. Jeantheau and C. Laredo. Parameter estimation for discretely observed stochastic volatility models. Bernoulli 5 (1999) 855-872. | MR | Zbl

A. Gloter. Discrete sampling of an integrated diffusion process and parameter estimation of the diffusion coefficient. ESAIM Probab. Statist. 4 (2000) 205-227. | Numdam | MR | Zbl

A. Gloter. Parameter estimation for a discrete sampling of an integrated Ornstein-Uhlenbeck process. Statistics 35 (2001) 225-243. | MR | Zbl

A. Gloter and J. Jacod. Diffusions with measurement errors. I. Local asymptotic normality. ESAIM Probab. Statist. 5 (2001) 225-242. | Numdam | MR | Zbl

E. Gobet. Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach. Bernoulli 7 (2001) 899-912. | MR | Zbl

E. Gobet. LAN property for ergodic diffusions with discrete observations. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 711-737. | EuDML | Numdam | MR | Zbl

F. Hirsch and S. Song. Criteria of positivity for the density of a Wiener functional. Bull. Sci. Math. 121 (1997) 261-273. | MR | Zbl

F. Hirsch and S. Song. Properties of the set of positivity for the density of a regular Wiener functional. Bull. Sci. Math. 122 (1998) 1-15. | MR | Zbl

I. A. Ibragimov and R. Z. Has'Minskii. Statistical Estimation. Asymptotic Theory. Springer-Verlag, New York-Berlin, 1981. (Translated from the Russian by Samuel Kotz.) | MR | Zbl

J. Jacod. On continuous conditional Gaussian martingales and stable convergence in law. In Séminaire de probabilité XXXI, 232-246. Lecture Notes in Math. 1655. Springer, Berlin, 1997. | EuDML | Numdam | MR | Zbl

P. Jeganathan. On the asymptotic theory of estimation when the limit of the log-likelihood ratios is mixed normal. Sankhyā Ser. A 44 (1982) 173-212. | MR | Zbl

P. Jeganathan. Some asymptotic properties of risk functions when the limit of the experiment is mixed normal. Sankhyā Ser. A 45 (1983) 66-87. | MR | Zbl

A. Kohatsu-Higa. Lower bounds for densities of uniformly elliptic random variables on Wiener space. Probab. Theory Related Fields 126 (2003) 421-457. | MR | Zbl

P. Krée and C. Soize. Mathematics of Random Phenomena. Random Vibrations of Mechanical Structures. D. Reidel Publishing Co., Reidel, Dordrecht, 1986. | MR | Zbl

H. Kunita. Stochastic differential equations and stochastic flows of diffeomorphisms. École d'été de probabilités de Saint-Flour, XII - 1982, 143-303. Lecture Notes in Math. 1097. Springer, Berlin, 1984. | MR | Zbl

L. Le Cam and G. Lo Yang. Asymptotics in Statistics. Some Basic Concepts, 2nd edition. Springer-Verlag, New York, 2000. | MR | Zbl

D. Nualart. The Malliavin Calculus and Related Topics. Probability and Its Application. Springer-Verlag, New-York, 1995. | MR | Zbl

D. Nualart. Analysis on Wiener space and anticipating stochastic calculus. Lectures on Probability Theory and Statistics (Saint-Flour, 1995), 123-227. Lecture Notes in Math. 1690. Springer, Berlin, 1998. | MR | Zbl

B. L. S. Prakasa Rao. Statistical Inference for Diffusion Type Processes. Edward Arnold, London; Oxford University Press, New York, 1999. | MR | Zbl

D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer-Verlag, Berlin, 1999. | MR | Zbl