Shape Restrictions, Empirical Processes, and Semiparametric Models

经验过程理论起源于如何从样本数据中估计概率分布的基本问题，现已发展成为一套强大的工具和统一的方法，用于研究各种统计过程的性质。这一研究项目旨在拓宽和深化这些工具，并开发其他方法。预计这项理论工作将在几个应用领域取得进展，包括流行病学和临床试验中使用的抽样设计，以及用于艾滋病毒/艾滋病数据的模型。该项目还涉及研究生的培训和研究生水平课程的开发。研究人员计划研究形状受限模型在单变量和多变量设置下的新统计方法。要研究的形状限制包括单调、凸函数、对数凹函数和对数凸函数。他的目标是开发新的基本经验过程工具和方法，并将新工具应用于涉及形状限制推理、半参数模型、自举推理方法和高维统计问题的问题。作者还打算介绍和评价基于按设计缺失数据的复杂抽样设计的半参数模型估计的新方法，以及形状约束定义的半参数模型估计的新方法。

Empirical process theory, which originated in the fundamental question of how to estimate a probability distribution from sample data, has developed into a powerful set of tools and unifying methods for the study of the properties of a wide range of statistical procedures. This research project aims to broaden and deepen these tools and to develop additional methods. It is anticipated that this theoretical work will enable progress in several application areas, including sampling designs used in epidemiology and clinical trials, as well as models used for HIV-AIDS data. The project also involves training of graduate students and development of graduate level courses. The investigator plans to study new statistical methods for shape-restricted models in univariate and multivariate settings. The shape restrictions to be studied include monotone, convex, log-concave, and log-convex functions. He aims to develop new basic empirical process tools and methods and to apply the new tools to problems involving shape-restricted inference, semiparametric models, bootstrap inference methods, and problems in high-dimensional statistics. The investigator intends also to introduce and evaluate new methods of estimation in semiparametric models based on complex sampling designs with missing data by design, and for estimation in semiparametric models defined by shape constraints.