Shape Restrictions, Empirical Processes, and Semiparametric Models

研究方向包括形状限制推理、经验处理方法和工具、半参数和非参数模型新理论等。特别是，研究者和华盛顿大学的一名研究生在凸函数估计和涉及形状限制的新半参数模型的背景下进行了新的置信集程序的研究。研究者和一名华盛顿大学生物统计学研究生进行了一项改进的估计方法的半参数模型与两阶段设计缺失数据（由设计）。本研究还涉及到各种复杂采样方法和多相设计的新渐近理论的发展。一些关于经验过程方法和工具的研究是与荷兰的同事共同进行的。一些形状约束估计下的推理研究是与法国、瑞士和加拿大的同事共同进行的。这些研究涉及极大似然估计量的非标准渐近性、似然比统计量和新的非标准极限分布。拟议研究的一部分涉及更好地理解高维环境下的自举和其他重采样程序。研究还包括开发基本的经验处理工具和方法，并将这些新工具和方法应用于半参数模型、形状限制模型和高维数据的统计问题。一个关键目标涉及改进经验似然理论和广义经验似然估计方法。另一个目标是了解重尾分布在高维统计问题中的影响。应用包括高维协变量的回归模型，缺少协变量数据的生存数据模型，以及用于艾滋病研究的非参数和半参数最大似然估计。两阶段数据依赖设计的工作可以应用于新的设计，提高了流行病学临床试验和病例队列抽样的效率。经验过程理论的工具允许对涉及高维数据和参数空间的其他统计领域中当前感兴趣的许多问题进行调查。这项研究通过培训研究生和将由此产生的新的统计方法纳入华盛顿大学统计和生物统计系的研究生课程，从而有利于教育和人类发展。

The investigator carries out research on problems involving shape restricted inference, empirical process methods and tools, and new theory for semiparametric and nonparametric models. In particular, the investigator and a University of Washington graduate student conduct research on new confidence set procedures in the context of convex function estimation and new semiparametric models involving shape restrictions. The investigator and a University of Washington graduate biostatistics graduate student conduct research on improved estimation methods for semiparametric models with two-phase designs with missing data (by design). This research also involves development of new asymptotic theory for a variety of complex sampling methods and multi-phase designs. Some of the research on empirical process methods and tools is carried out jointly with colleagues in the Netherlands. Some of the research on inference under shape constrained estimation is carried out jointly with colleagues in France, Switzerland, and Canada. These investigations involve nonstandard asymptotics for maximum likelihood estimators, likelihood ratio statistics, and new nonstandard limit distributions. Part of the proposed research involves better understanding of bootstrap and other resampling procedures in high-dimensional settings. The research also involves development of basic empirical process tools and methods, and applications of these new tools and methods to statistical problems concerning semiparametric models, shape restricted models, and high-dimensional data. One key goal involves improved theory for empirical likelihood and generalized empirical likelihood estimation methods. Another goal is to understand the effect of heavier tailed distributions in high-dimensional statistical problems. Applications include regression models with high-dimensional covariates, models for survival data with missing covariate data, and non- and semi-parametric maximum likelihood estimators used in HIV-AIDS research. The work on two-phase data dependent designs has application to new designs with increased efficiency for clinical trials and case-cohort sampling in epidemiology. The tools of empirical process theory allow investigations of many problems of current interest in other areas of statistics involving high-dimensional data and parameter spaces. The research benefits education and human development by the training of graduate students and the inclusion of the resulting new statistical methods in graduate level courses for the Departments of Statistics and Biostatistics at the University of Washington.