Adaptive nonparametric regression - a nonasymptotic approach

当未知函数的观测值包含随机误差时，恢复该函数的问题称为回归。如果函数不能用有限数量的参数来描述，那么它就是非参数回归。在这项提议中，我们追求这一理论未来成功所必需的几个目标。 第一个目标是结合两个成熟的统计估计理论：最优设计和非参数回归。这两种理论都成功地发展了半个世纪，彼此之间几乎没有什么影响，尽管两种理论都有一个共同的模式：回归。在实际应用中，必须将这两个领域的思想和方法结合起来，形成一个统一的理论。 第二个目标是进一步发展最优非参数回归的非渐近方法。现代功能强大的计算机不仅可以帮助比较不同的数字估计技术，而且还可以更好地理解纯粹渐近方法的限制，这种方法在很大程度上主导了过去的统计理论。 第三个目标是将作者最近在非参数回归的传统背景下发展的非渐近最优方法扩展到更具挑战性的自适应或数据驱动的估计方法。该项目的这一部分与众所周知的变点问题和“Oracle不等式”密切相关。 该项目的一个重要部分是培训HQP，他们将具备未来发展非参数统计所需的工具，这些工具在数据分析、信号处理、生物统计、化学工程和实验物理等领域很有用。

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.