High-dimensional statistical inference in parametric and nonparametric models

技术、工程和计算能力的最新进展，以及基因组学、临床试验、宇宙学和气候研究等不同领域的问题，产生了涉及大量未知参数的新型推理问题，这些问题被称为高维推理或稀疏推理问题。一般来说，很难完全非参数化地处理这些问题并提供具有几乎精确理论性质的程序。研究计划的目的是发展新的和改进现有的一些高维统计的推理程序。在这样做的过程中，重点是在诸如估计理论、假设检验理论、变量选择和分类等数学统计领域提供最佳和自适应（不需要了解统计模型的未知参数）程序。采用的主要方法是非参数的。这种方法被世界范围内的数理统计学家所采用，它假定进入所研究的统计模型的参数是无限维的。例如，在非参数回归分析中，混合了弱噪声的未知回归函数被假设为一个大（无限维）函数类的成员，而不是依赖于有限数量的未知参数的已知函数，如在参数回归分析中。本研究采用的统计程序的主要优良标准是渐近极小。这种强烈的最优性概念通常用于现代非参数统计推断。我们期望在完成本研究计划期间开发的方法将在临床试验、天体物理学、经济学和信息技术等不同领域得到应用。

Recent advances in technology, engineering, and computing power, as well as problems in diverse areas such as genomics, clinical trials, cosmology, and climate studies have given rise to new types of inference problems that involve a very large number of unknown parameters and are known as high-dimensional inference or sparse inference problems. In general, it is difficult to treat these problems fully nonparametrically and provide procedures with nearly exact theoretical properties. The aim of the research proposal is to develop new and improve some existing inferential procedures of high-dimensional statistics. In doing so, the emphasis is on providing optimal and adaptive (not requiring the knowledge of unknown parameters of the statistical models) procedures in such areas of mathematical statistics as estimation theory, hypothesis testing theory, variable selection and classification. The main approach to be taken is nonparametric. This approach, adopted by mathematical statisticians on a worldwide scale, assumes that the parameter(s) entering the statistical models under study are infinite-dimensional. For instance, in nonparametric regression analysis, an unknown regression function mixed with weak noise is assumed to be a member of a large (infinite-dimensional) class of functions, rather than a known function depending on a finite number of unknown parameters, as in parametric regression analysis. The main criteria of goodness of a statistical procedure employed in this study is asymptotic minimaxity. This strong notion of optimality is commonly used in modern nonparametric statistical inference. We anticipate that the methods developed during the completion of this research proposal will find their usage in diverse fields such as clinical trials, astrophysics, economics, and information technology.