Adaptive nonparametric regression - a nonasymptotic approach

自适应非参数回归 - 一种非渐近方法

• 批准号: 238442-2010
• 负责人: Levit, Boris
• 金额: $ 1.09万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570422
• 关键词: Adaptive; nonparametric; regression; nonasymptotic; approach

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.

当未知函数的观测值包含随机误差时，恢复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，则它是非参数回归。在这个建议中，我们追求这个理论未来成功所必需的几个目标。 第一个目标是结合联合收割机两个完善的统计估计理论：最优设计和非参数回归。这两种理论已经成功地发展了半个世纪，尽管两者都有一个共同的模型：回归，但彼此影响甚微。为了实际应用，必须将这两个领域的思想和方法联合收割机整合成一个统一的理论。 第二个目标是进一步发展一个非渐近的方法，以最佳的非参数回归。现代强大的计算机不仅可以帮助比较数值上不同的估计技术，而且还可以更好地理解纯粹渐近方法的限制，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传统的非参数回归环境中开发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。这一部分的项目是密切相关的著名的变点问题和“甲骨文不等式。" 该项目的一个重要组成部分是HQP的培训，谁将配备所需的工具，为非参数统计的未来发展，有用的数据分析，信号处理，生物统计，化学工程和实验物理等领域。