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.

Abstract (VO)

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 $X$.

Résumé

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.

Détail

DOI : 10.1214/07-AIHP111

Classification : 60F99, 60H07, 62F12, 62M09
Keywords: diffusion processes, parametric estimation, LAMN property, Malliavin calculus, non-markovian data

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