Mathematical Sciences: Some Problems for Incomplete Survival Data

本项目处理生存数据的三个一般统计问题，这些数据要么不完整，要么受到选择偏差的影响。为了演示简单，我们将重点放在右删节/左截断的数据上。这种不完整的数据可能出现在延迟进入的随访研究中。第一个问题处理基于不完全数据或带有选择偏差的数据的产品极限估计泛函的一些基本性质，如强大数定律（SLLN）和中心极限定理（CLT）。除了经过审查的数据外，这些基本结果到目前为止还没有，将进行调查。SLLN和CLT在统计推断方面的应用非常广泛，也有待进一步研究。第二个问题处理不完全数据的m估计量。对于不完全数据，即使对于右截尾数据，也没有处理m估计量极限理论的一般方法。我们打算给出基于不完全数据的m -估计的强相合性和渐近正态性的一般解析性充分条件。对于不完全数据的m估计的鲁棒性问题也将被探讨。第三个问题处理不完整数据的降维方法。这些方法并没有在文献中流行起来，因为不完整的数据，维度的诅咒比广泛探索的未经审查的情况要严重得多。我们特别关注最近有前途的方法，切片逆回归（SIR）。程序的稳健性和对给定协变量（如回归函数）的响应变量估计统计量的一些问题将包括在内。该项目处理终身数据的一般统计问题，例如，某种机械或电子产品的寿命或艾滋病等疾病的潜伏期。寿命数据的一个共同特点是很难观测到某些实际寿命，因此这些观测结果在统计上被称为“不完全”。不完全数据的形式多种多样，其中最常见的是“删节”和“截断”。我们的研究重点是，但不限于，这些类型的不完整的数据。在理想的情况下，所有数据都可以被充分观察到，大多数感兴趣的统计量，如生存概率或某些疾病的风险，都可以通过经验来估计。与这种情况相比，数据的不完整性带来了非常具有挑战性的统计问题，许多基本性质或结构仍未解决或未知。在这个项目中，将研究三个不完整数据的具体开放问题。第一部分处理两个最基本的概率性质，强大数定律和不完全数据的中心极限定理。这些性质在概率论中起着核心作用，对统计推断是必不可少的。然而，直到最近，研究人员才能够接触到一些特殊类型的不完整数据。我们的目标是为其他一般类型的不完整数据建立这样的基本结果。项目这一部分的研究结果将有助于研究不完整数据的可靠统计程序，例如m估计器。本课题要探讨的第三个问题是响应变量可能不完全时的高维数据分析方法。即使对于完整的数据，通过降维方法处理高维数据也需要特殊的技能，并且处于统计研究的前沿。提出的处理不完整数据的程序扩展了现有降维程序的范围和有用性。

This project deals with three general statistical problems for survival data which are either incomplete or subjected to selection bias. For demonstration simplicity we focus on right censored/left truncated data. Such incomplete data may arise in follow-up studies with delayed entries. The first problem deals with some basic properties like the strong law of large numbers (SLLN) and the central limit theorem (CLT) for functionals of product-limit estimates based on incomplete data or data with selection bias. Except for censored data such fundamental results are not available so far and will be investigated. Applications of SLLN and CLT are plentiful in statistical inference and will also be studied. The second problem deals wth M-estimators for incomplete data. General approach to handle the limit theory of M-estimators is not yet available for incomplete data even for right censored data. We intent to develop general analytical sufficient conditions for the strong consistency and asymptotic normality of M-estimators based on incomplete data. Robustness issues of M-estimators for incomplete data will also be explored. The third problem deals with dimension reduction methods for incomplete data. Such methods have not caught on in the literature for incomplete data where the curse of dimensionality is much more serious than the widely explored noncensored case. We focus in particular on a recent promising method, sliced inverse regression (SIR). Some issues on robustness of the procedure and estimating statistical quantities of the response variable for a given covariate, such as the regression function, will be included. The project deals with general statistical problems for lifetime data, for example, the life time of a certain mechanical or electronical product or the incubation time of a disease such as AIDS. One common feature of lifetime data is the difficulties of observing some of the actual lifetimes and those observations are thus termed "incomplete" sta tistically. Incomplete data arise in various forms among which "censoring" and "truncation" are the most common ones. Our study focus on, but not limited to, those types of incomplete data. In the idealistic situation where all data can be observed fully most statistical quantity of interest, such as the survival probability or risk of certain disease, can be estimated empirically. Compared to such a situation the incompleteness of the data poses very challenging statistical problems and many of the basic properties or structures remain unsolved or unknown. In this project, three specific open problems for incomplete data will be investigated. The first one deals with the two most fundamental probability properties, the strong law of large numbers and central limit theorem for incomplete data. Such properties play central role in probability theory and are essential for statistical inferences. However, it is only until very recently that researachers are able to put their hands on some special type of incomplete data. Our goal is to establish such fundamental results for other general type of incomplete data. The findings in this part of the project will facilitate the study of robust statistical procedures, such as M-estimators, for incomplete data. The third problem to be explored in this project deals with high dimensional data analytical methods when the response variable are possibility incomplete. Even for complete data, the handling of high dimensional data, via dimensional reduction methods, requires special skills and is on the cutting edge of statistical research. The proposed procedure for handling incomplete data extends the scope and usefullness of existing dimension reduction procedures.