Mathematical Sciences: Estimation in Semiparametric Models and Empirical Processes

DMS 95-32039：Wellner本研究涉及经验过程、统计学中的Bootstrap方法、半参数模型、反问题和非标准渐近性的研究。研究内容包括无穷维M-估计的极限理论和相关的Bootstrap方法，一致Donsker函数类的保持定理，一致P Bootstrap极限定理，单调函数和凸函数的估计，迭代凸极小算法的收敛，以及区间截尾模型中整体泛函的行为。%对于受右截断和/或左截断影响的故障时间数据的分析，现在有许多方法可用。然而，在许多生物医学研究中，包括那些来自艾滋病毒/艾滋病研究的研究，失败时间被以更复杂的方式审查或截断。一种这样的并发症是“间隔审查”，在这种情况下，人们知道感兴趣的事件发生在随机的时间间隔内，但不知道确切的时间：例如，艾滋病毒阳性或CD4计数下降到某一阈值以下可能只被确定为在两次临床就诊之间，但确切的时间可能仍然未知。这项研究将涉及为这些和其他更复杂的审查机制产生的数据开发新的统计方法，范围从动物致癌实验到社会学和计量经济学模型中的数据。将完成对其他统计学家最近提出的方法的性质的详细研究和评价。***

DMS 95-32039: Wellner This research involves the study of empirical processes, bootstrap methods in statistics, semiparametric models, and inverse problems and nonstandard asymptotics. The research will involve limit theory for infinite - dimensional M - estimates and related bootstrap methods, preservation theorems for uniform Donsker classes of functions, uniform in P bootstrap limit theorems, estimation of monotone and convex functions, convergence of iterative convex minorant algorithms, and the behavior of global functionals in models with interval censoring. %%% For the analysis of failure time data which are subject to right censoring and/or left truncation, many methods are now available. However, in many biomedical studies, including those arising from HIV/AIDS studies, failure times are censored or truncated in more complicated ways. One such complication is ``interval censoring'', in which one knows that the event of interest occurs within a random time interval, but not the exact timing: for example, HIV positivity or the drop of the CD4 count below a certain threshold may only be determined to be between two clinical visits, but not the exact time may remain unknown. This research will involve developing new statistical method for data arising from these and other more complicated censoring mechanisms ranging from animal carcinogenesis experiments to data in sociology and econometrics modelling. A detailed study and evaluation of the properties of methods recently proposed by other statisticians will be completed. ***