Theory for General Regression with Heavy Tails and Shape Constraints: A Multiplier Empirical Process Approach

现代高维复杂结构数据集通常存在噪声和重尾现象。该项目将解决由于这些高度噪声的数据集和不稳定的调整敏感的非参数方法所带来的挑战而产生的两个主要的相互关联的问题。第一个问题涉及高维复杂结构数据集的噪声性质。它阐述了我们可以在多大程度上相信为误差非常小的回归模型设计的估计程序的结果，同时记住，由于大量的异常值，这种理想的假设在实践中可能会悲惨地失败。第二个问题涉及微妙且通常不稳定的调谐敏感多维非参数方法。它的目的是了解具有附加形状约束的替代的免调谐和稳定的统计估计方法在多大程度上是可靠的。研究生将参与开发多维形状约束模型的理论和方法，实现算法，进行数值实验，并验证统计方法的理论属性。拟议的研究问题在乘数经验过程(经验过程的一种特殊形式)表现出的潜在概率结构水平上有密切的共同联系。不幸的是，由于流程内部的复杂结构，现有工具必然无法提供敏锐的理解。PI将开发与这些乘数经验过程相关的概率工具和技术。这些工具和技术将在理解具有重尾和形状约束的回归模型中常用的最小二乘法和其他相关的频域和贝叶斯方法的相变行为方面发挥关键作用。正在研究的特殊问题包括(A)具有重尾误差的非参数模型中惩罚最小二乘估计器的行为，(B)具有重尾误差的准高斯似然下的贝叶斯过程的行为，(C)多维保序和凸形约束模型中最小二乘估计器的行为，以及(D)具有形状约束的一般加性模型中的最大似然估计器的行为。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Modern high-dimensional complex-structured datasets are usually very noisy and heavy-tailed. The project will address two major inter-related questions arising from the challenges due to these highly noisy datasets and unstable tuning-sensitive nonparametric methods. The first question concerns the noisy nature of high-dimensional complex-structured datasets. It addresses the extent to which we can believe in the results of estimation procedures designed for regression models with very light-tailed errors, while keeping in mind that such an ideal assumption may fail miserably in practice due to numerous outliers. The second question concerns the subtle and usually unstable tuning-sensitive multidimensional nonparametric methods. It aims at understanding how far alternative tuning-free and stable statistical estimation methods with additional shape constraints can be reliable. The graduate student will be involved in developing theory and methods for multi-dimensional shape constrained models, implementing algorithms, performing numerical experiments, and validating the theoretical properties of the statistical methods.The proposed research questions share a close common tie at the level of their underlying probabilistic structures exhibited by multiplier empirical processes, a special form of the empirical processes. Unfortunately, existing tools necessarily fail to provide sharp understandings due to complex structures within the processes. The PI will develop probabilistic tools and techniques in connection with these multiplier empirical processes. These tools and techniques will play a crucial role in understanding the phase-transitional behavior of the commonly used least squares and other related frequentist and Bayes methods in regression models with heavy tails and shape constraints. The particular problems under investigation include (a) behavior of penalized least squares estimators in nonparametric models with heavy-tailed errors, (b) behavior of Bayes procedures under quasi-Gaussian likelihood with heavy-tailed errors, (c) behavior of least squares estimators in multi-dimensional isotonic and convex shape constrained models and (d) behavior of maximum likelihood estimators in general additive models with shape constraints.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.