On estimating the diffusion coefficient: Parametric versus nonparametric

We consider the following problem: estimate the Lipschitz continuous diffus ion coefficient sigma (2) from the path of a 1-dimensional diffusion proces s sampled at times i/n, i = 0,..., n, when we believe that sigma (2) actual ly belongs to a smaller regular parametric-set Co. By introducing random no rmalizing factors in the risk function, we obtain confidence sets which can be essentially better than the minimax rate n(-1/3) of estimation for Lips chitz functions in diffusion models. With a prescribed confidence level alp ha (n), we show that the best possible attainable (random) rate is (root lo g alpha (-1)(n) /n)(2/5). We construct an optimal estimator and an optimal random normalizing factor in the sense of Lepski (1999). This has some consequences for classical estimation: our procedure is adapt ive w.r.t. Sigma (0) and enables us to test the hypothesis that sigma (2) i s parametric against a family of local alternatives with prescribed 1st and 2nd-type error probabilities, (C) 2001 Editions scientifiques et medicales Elsevier SAS.