Adaptive nonparametric regression - a nonasymptotic approach

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When observations of an unknown function contain random errors, the problem of recovering the function is known as Regression. If the function cannot be described by a finite number of parameters, it is a Nonparametric Regression. In this proposal, we pursue several goals essential for the future success of this theory. The first goal is to combine two well established theories of Statistical Estimation: the Optimal Design and Nonparametric Regression. Both theories have been successfully developing for half a century, having little effect on one another, despite the fact that both have a common model in sight: Regression. For practical applications, it is essential to combine the ideas and methods of both fields into a unified theory. The second goal is to develop further a non-asymptotic approach to optimal Nonparametric Regression. Modern powerful computers can not only assist in comparing numerically different estimation techniques, but also in better understanding the restrictions of the purely asymptotic approach which, in a significant part, dominated statistical theory of the past. The third goal is to extend the nonasymptotic optimality approach developed recently by the author in a traditional setting of Nonparametric Regression, to the more challenging adaptive, or data-driven, methods of estimation. This part of the project is closely related to the well known Change-Point problem and "Oracle Inequalities." An essential part of this project is the training of HQP, who will be well equipped with the tools required for the future development of nonparametric statistics, useful in such fields as data analysis, signal processing, biostatistics, chemical engineering, and experimental physics.