Mathematical Sciences: Some Problems for Incomplete Survival Data

本项目主要研究不完全或存在选择偏差的生存数据的三个一般统计问题。为了演示简单，我们专注于右删失/左截断数据。此类不完整数据可能出现在延迟录入的随访研究中。第一个问题涉及一些基本性质，如强大数定律（SLLN）和中心极限定理（CLT）的泛函的乘积极限估计的基础上不完全数据或数据的选择偏差。除了删失数据，这些基本结果到目前为止还没有，将进行调查。SLLN和CLT在统计推断中的应用是丰富的，也将被研究。第二个问题是不完全数据的M-估计。对于不完全数据，即使是右删失数据，M-估计的极限理论的一般处理方法也是不存在的。本文研究了不完全数据下M-估计的强相合性和渐近正态性的一般分析充分条件。还将探讨不完全数据的M-估计量的稳健性问题。 第三个问题涉及不完全数据的降维方法。这样的方法还没有赶上在文献中的不完整数据的维数灾难是更严重的比广泛探讨的非删失的情况下。 我们特别关注最近有前途的方法，切片逆回归（SIR）。将包括关于程序的稳健性和估计给定协变量（如回归函数）的响应变量的统计量的一些问题。该项目涉及寿命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征是难以观测某些实际寿命，因此这些观测在统计学上被称为“不完整”。 不完全数据以各种形式出现，其中“删失”和“截断”是最常见的形式。我们的研究重点是但不限于这些类型的不完整数据。在理想的情况下，所有的数据都可以完全观察到，大多数感兴趣的统计量，如生存概率或某些疾病的风险，可以根据经验估计。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然没有解决或未知。在这个项目中，三个具体的开放问题，不完整的数据将进行调查。第一个涉及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发挥着核心作用，对于统计推断也至关重要。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。我们的目标是建立其他一般类型的不完全数据的基本结果。本项目这一部分的研究结果将有助于研究不完整数据的稳健统计程序，如M估计量。本计画探讨的第三个问题是当回应变数可能性不完全时的高维资料分析方法。即使是完整的数据，高维数据的处理，通过降维方法，需要特殊的技能，是统计研究的前沿。处理不完整数据的建议程序扩展了现有的降维程序的范围和实用性。

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.