CAREER: Nonparametric function estimation: shape constraints, adaptation, inference and beyond

非参数统计是统计学和机器学习的一个领域，它允许人们在不对数据做出强有力的事先假设的情况下对数据集进行建模和分析。非参数统计技术有用的数据问题来自广泛的应用领域，包括生物学、医学、天文学、工程学、经济学和运筹学。在现代复杂和大型数据集中，这些方法尤其重要，因为它们能够检测数据中的重要趋势和模式，而传统的参数统计技术可能会遗漏这些趋势和模式。然而，在现代数据问题中，非参数方法的理论、方法和应用还存在许多悬而未决的问题。本项目将对这些问题进行系统研究，其结果是：(A)提高对许多现有方法的理解(在精确度和不确定性量化方面)，以及(B)将对上述科学领域的应用从业者有用的新方法和计算算法。大多数拟议的项目都是合作的，涉及来自不同大学的研究人员。该项目还包含一项完善的教育活动计划，这将对加州大学伯克利分校本科生和研究生在统计研究方面的教育和培训产生重大影响。特别是，该项目的许多教育活动都是针对本科生的，这一群体在大型研究型大学中往往不那么重要。具体来说，将研究广泛的非参数模型，包括回归和密度估计。在存在经验吸引估计量的情况下，提出了一种详细的理论研究，重点研究了它们的自适应风险性质。在其他情况下，提出了估计器和有效的计算算法，并分析了它们的精度。文中还讨论了推理和不确定性量化的重要实际问题。本项目研究的具体回归问题包括：(A)多元凸回归、单变量趋势滤波和加性形状约束回归，其中自然估计的自适应风险性质将被建立；(B)多元趋势滤波和拟凸回归，其中提供新的估计量和有效的计算算法；以及(C)形状约束估计中的全局和逐点推理，其中不确定性将被量化。在密度估计方面，所研究的问题包括：(A)对数凹和混合密度估计，其中将研究最大似然估计；(B)分布稳健优化和非高斯分量分析，其中将提出基于形状约束密度估计的新方法；以及(C)将开发新程序的形状约束推断的稳健方法。

Nonparametric statistics is an area of statistics and machine learning that allows one to model and analyze datasets without making strong prior assumptions about the data. Data problems where the techniques of nonparametric statistics are useful come from a wide variety of applied areas including biology, medicine, astronomy, engineering, economics and operations research. In modern complex and large datasets, these methods are especially crucial as they enable the detection of important trends and patterns in the data that may be missed by traditional parametric statistical techniques. However there exist many unresolved issues concerning the theory, methodology and application of nonparametric methods in modern data problems. A systematic study of these issues will be undertaken in this project which will result in (a) an improved understanding (in terms of accuracy and uncertainty quantification) of many existing methods, and (b) novel methods and computational algorithms that will be useful to applied practitioners in the scientific areas mentioned above. Most of the proposed projects are collaborative and involve researchers from a diverse set of universities. The project also contains a well-developed plan of educational activities which will have a major impact on the education and training of undergraduate and graduate students at UC Berkeley in statistical research. In particular, many of the educational activities of the project are aimed towards undergraduate students, a group that is often given less importance at large research universities.Concretely, a wide range of nonparametric models will be studied, covering both regression and density estimation. In situations where empirically attractive estimators exist, an elaborate theoretical study is proposed focusing on their adaptive risk properties. In other situations, estimators and efficient computational algorithms are proposed together with an analysis of their accuracy. Important practical problems of inference and uncertainty quantification are also addressed. The specific regression problems that are investigated in this project include (a) multivariate convex regression, univariate trend filtering and additive shape constrained regression where adaptive risk properties of the natural estimators will be established, (b) multivariate trend filtering and quasi-convex regression where new estimators are provided along with efficient computational algorithms, and (c) global and pointwise inference in shape constrained estimation where uncertainty quantification will be addressed. In density estimation, the problems investigated include: (a) log-concave and mixture density estimation where maximum likelihood estimators will be studied, (b) distributionally robust optimization and nongaussian component analysis where novel methodology will be proposed based on shape-constrained density estimation, and (c) robust approaches to shape-constrained inference where new procedures will be developed.