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.
