High-dimensional statistical inference in parametric and nonparametric models

最近技术、工程和计算能力的进步，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，被称为高维推理或稀疏推理问题。一般来说，很难完全非参数地处理这些问题，并提供具有几乎精确的理论性质的程序。研究建议的目的是开发新的和改进现有的高维统计的一些推理程序。在这样做的过程中，重点是在估计理论、假设检验理论、变量选择和分类等数理统计领域提供最佳和适应的程序(不需要知道统计模型的未知参数)。要采取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采用，假设进入所研究的统计模型的参数(S)是无限维的。例如，在非参数回归分析中，假设混合有弱噪声的未知回归函数是大(无限维)函数类的成员，而不是像在参数回归分析中那样依赖于有限数量的未知参数的已知函数。本研究中使用的统计过程优劣的主要标准是渐近极小公理。这种最优性的强烈概念在现代非参数统计推断中被普遍使用。我们预计，在完成本研究提案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.