Shape Restrictions, Semiparametric Models, and Empirical Processes

Jon a . Wellner将对各种形状限制、审查数据和缺失数据问题的半参数和非参数模型的经验过程方法和计算策略进行研究。特别是，Wellner和华盛顿大学的研究生Marios Pavlides将对几个模型进行研究，这些模型涉及多元单调密度函数和多元当前状态观察方案，这是目前与艾滋病毒/艾滋病研究相关的一类模型。Wellner和华盛顿大学的研究生Arseni Seregin将研究对数凹密度的非参数估计及其在一维和高维上的推广。一些关于经验过程的研究将与Aad van der Vaart（阿姆斯特丹自由大学）联合进行。部分形状约束下的推理研究将与Fadoua Balabdaoui、Marloes Maathuis、Shuguang Song（原博士生）共同开展。新的计算算法的工作将与Hanna Jankowski共同进行，Hanna是华盛顿大学最近的博士后学生，现在是加拿大约克大学的教职员工。这些研究将涉及极大似然估计的非标准渐近性，似然比统计，以及新的非标准极限分布的分布理论。研究将包括开发基本的经验过程工具和方法，并将这些新工具和方法应用于半参数模型、形状限制模型和反问题的统计问题。应用包括面板计数数据的回归模型，几种类型的双变量区间剔除数据，缺少协变量数据的生存数据的回归模型，艾滋病研究中使用的非参数和半参数最大似然估计的研究，形状限制推断的新置信带，以及基本经验过程理论和两阶段数据相关设计的新半参数估计程序。竞争风险模型推理方法的进展被用于研究HIV疫苗试验中病毒载量的因果效应。双变量审查数据的发展应用于艾滋病毒-艾滋病母婴传播的研究。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和其他地方的统计课程，从而有利于教育和人类发展。

Jon A. Wellner will carry out research on empirical process methods and computational strategies for a variety of semiparametric and nonparametric models for shape restrictions, censored data, and missing data problems.In particular, Wellner and University of Washington graduate student Marios Pavlides will conduct research on several models involving multivariate monotone density functions and multivariate current status observation schemes, a class of models currently of interest in connection with HIV-AIDS studies. Wellner and University of Washington graduate student Arseni Seregin will conduct research on nonparametric estimation of log-concave densities and generalizations thereof in one and higher dimensions. Some of the research on empirical processes will be carried out jointly with Aad van der Vaart (Free University, Amsterdam). Some of the research on inference under shape constraints will be carried out jointly with Fadoua Balabdaoui, Marloes Maathuis, and Shuguang Song (former Ph.D. students). The work on new computational algorithms will be carried out jointly with Hanna Jankowski, a recent post-doctoral student at the University of Washington and now a faculty member at York University, Canada. These investigations will involve nonstandard asymptotics for maximum likelihood estimators, likelihood ratio statistics, and distribution theory for new nonstandard limit distributions.The research will involve 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 inverse problems. Applications include regression models for panel count data, bivariate interval censored data of several kinds, regression models for survival data with missing covariate data, studies of non- and semi-parametric maximum likelihood estimators used in AIDS research, new confidence bands for shape restricted inference, and both basic empirical process theory and new semiparametric estimation procedures for two-phase data dependent designs. The advances in inference methods for competing risks models are used to study causative effects of viral loading in HIV vaccine trials. The developments in bivariate censored data are applied to studies of mother-to-child transmission of HIV-AIDS. 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 statistics courses at the University of Washington and elsewhere.