Abstract:  The talk will be in three parts. The first part is a gentle introduction and review of some easy-to-understand geometric concepts like tangent space and canonical gradient. These concepts are used in describing asymptotic information bounds in parametric, non-parametric and semi-parametric estimation when using independent and identically distributed observations. The same concepts are more difficult to define when the observations are dependent. We will outline a suitable definition and a generalization for the case of stationary geometrically egodic Markov chains. The concepts will be applied to formulate some efficiency results for nonparametric estimation of distribution functions and of nonparametric functionals.

In the second part, a nonparametric functional estimation problem for Markov chains with a parametric restriction on the marginal will be outlined. Again, the efficient estimation issue will be discussed. An explicit construction of an efficient estimator of a functional of two consecutive observations of the chain under parametric restriction on the marginal will be outlined.

The problem discussed in the second part, has a specific application in estimation of discretely observed ergodic diffusions where often the parametric form of the marginal is known quite precisely but the dynamics may not be very well known. In the third part, numerical examples and simulations related to a simple discretely observed ergodic diffusion model will be presented.