In this thesis we study adaptive methods of estimation for two particular
types of statistical problems: regression and density estimation. For all
these problems the classes of probabilities are parameterized by real-valued
functions. In each model, the underlying function is assumed to belong to
some class of smooth functions. In practice the `true' smoothness of the
function is unknown and so the actual class is also unknown.
We study different regression problems with fixed discrete designs:
regression on the real line and regression on a bounded
interval. Formally, the distinction here lies just in the definition of the
underlying functional classes. The construction of optimal adaptive
procedures however is quite different in these cases. This is underlined by
the essential difference between these two models; namely, in the case of
regression models on bounded observation intervals, the presence of the
boundary -- the so called boundary effect -- has to be incorporated in the
study of optimal statistical procedures.
For each of the three problems: regression on the real line, regression on
bounded intervals and density estimation, we introduce corresponding scales
of functional classes for which exact -- up to constants -- rates of
convergence are obtained, under the classical minimax non-parametric
framework, i.e. in the case when the classes are known. We proceed then by
constructing adaptive estimators and prove them to be asymptotically
optimal, for the corresponding functional scales