• Cox process
• Cramer-Rao lower bound
• drift estimation
• intensity estimation
• Malliavin calculus
• score function
• Stein estimator

The purpose of this thesis is to investigate the use of Malliavin Calculus in both parametric and nonparametric statistical inference. It is of interest to see how classical statistical results, which rely on the integration by parts formula, can be established in more general settings using Malliavin Calculus techniques. This study essentially consists of two parts: the probabilistic theory of Malliavin Calculus and its applications in statistics.

We first provided a collection of the main results concerning Malliavin Calculus on the canonical Wiener space, using the stochastic calculus of variations approach. It is worth mentioning the integration by parts formula, which plays a major role in this context, the Clark-Ocone-Karatzas formula and the derivability in Malliavin sense of the solutions of stochastic differential equations.

There are several books and papers which extend the classical Malliavin Calculus to the Poisson space. Using the experience with the Brownian motion and the Poisson process, we introduced Malliavin Calculus for doubly stochastic processes, following again the stochastic calculus of variations approach. In particular we showed that the properties we were interested in, such as the integration by parts formula, the chain rule and an explicit representation of the divergence for deterministic processes, are still valid.

Once we had established the basics of Malliavin Calculus, we turned our attention towards exploring its applications in a parametric statistical model relying on a recent paper by José M. Corcuera and A. Kohatsu-Higa. Making use of Malliavin Calculus we derived expressions of the score function as a conditional expectation involving Skorohod integral. Then one can immediately obtain the Fisher Information and the Cramer-Rao lower bound. In most classical models the calculations are straightforward, as the expression of the density is available. The goal was to show that in some cases we can derive such expressions without knowing the likelihood explicitly. In particular we find out that this method is appropriate to study asymptotic properties of continuous time models considering discrete observations of diffusion processes where the driving process is a Brownian motion.

Finally, the last part relies on some results established by N. Privault and A. Réveillac but we focused on two particular problems of nonparametric functional estimation: drift estimation for the Brownian motion and intensity estimation for the Cox process. We provided Cramer-Rao bounds and extended Stein's argument for superefficient estimators to an infinite dimensional setting using Malliavin Calculus. In addition we discussed the estimation in fractional Sobolev spaces of the unknown function, which is assumed to belong in the space H^1_0. We argued that, although it would be natural to look for an unbiased estimator which belongs to the same space as the target function, its risk will always be infinite. We also distinguished between the spaces W^{\alpha,2}, where the observation itself turns out to be an efficient estimator and the spaces W^{\alpha,p}, p\in(2,\infty), where it is not.

In conclusion, Malliavin Calculus provides a useful instrument for giving alternative expressions of the score function without involving the likelihood function directly. Consequently we obtain the Fisher Information, the Cramer-Rao lower bound and study the asymptotic behaviour of the model. On the other side it enables us to apply Stein's argument in the context of functional estimation.