Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### On estimating the diffusion coefficient: parametric versus nonparametric

Auteur(s):

Code(s) de Classification MSC:

• 62G05 Estimation
• 62G15 Tolerance and confidence regions
• 62G20 Asymptotic properties
• 62M02 Markov processes: hypothesis testing
• 62M05 Markov processes: estimation

Résumé: We consider the following problem: estimate the Lipschitz continuous diffusion coefficient $\sig$ from the path of a 1-dimensional diffusion process sampled at times $i/n, i=0,\ldots,n$, when we believe that $\sig$ actually belongs to a smaller regular parametric set $\Sigma_0$. By introducing random normalizing factors in the risk function, we obtain confidence sets which are essentially better than the $n^{-1/3}$ rate of estimation of Lipschitz functions. With a prescribed confidence level $\alpha_n$, we show that the best possible attainable (random) rate is $\left(\sqrt{\log \alpha_n^{-1}}/n\right)^{2/5}$. We construct an optimal estimator and an optimal random normalizing factor in the sense of Lepski \cite{LEPSKI}. This has some consequences for classical estimation: our procedure is adaptive w.r.t. $\Sigma_0$ and enables us to test the hypothesis that $\sig$ is parametric against a family of local alternatives with prescribed 1st and 2nd-type error probabilities.

Mots Clés: Random normalizing factors ; minimax theory ; adaptive estimation ; nonparametric testing ; diffusion processes

Date: 1999-09-02

Prépublication numéro: PMA-521