| This dissertation research has been oriented by the goal of establishing the theoretical foundation of sharp-adaptation, which can be outlined in the sense of the questions as follow. (1) What is the nonparametric limit of estimating a curve, or recovering a signal from noise, under suitable smoothness regularization of the curve? (2) How do we break such a limit by structural assumptions, compatibly built over the classical minimax paradigm, on the properties of a curve in nonparametric estimation? (3) Can we characterize smoothness properties or patterns of a curve with feasible structural assumptions under general regularizations and estimate such characteristics sharply? (4) How confident are we about our assumptions and the estimates under them? In this article, we present some fundamental results and clues to these questions in the framework of a simple and highly extensible methodology for nonparametric curve estimation. |
