Statistical Methods for Functional Data and Failure Time Data

功能数据和故障时间数据在生物学、医学、生物医学工程、环境工程和生物科学等许多科学领域中经常出现。我提出的研究计划将主要集中在开发一系列统计方法和策略来系统地分析这些数据，然后解释收集数据的受试者的行为。近年来，功能数据分析（FDA）引起了许多研究者的关注，并发展了几种方法。现有方法的关键是用低维变量表示高维数据，而低维变量可以用多元随机变量的方法进行分析。然而，如果观察的维度是超高的，将多变量技术直接推广到功能分析领域通常是不可行的。因此，我计划从不同的角度关注FDA，并为这一研究领域开发一些方法。我还将开发分析多元区间截尾数据的方法。我将考虑对具有审查机制的医疗费用数据进行联合建模，并对临床试验的纵向有序数据进行分析。实值和巴拿赫空间值随机变量的自归一化和的极限定理一直是我的研究兴趣。我将得到自归一化和的结果，如迭代对数定律、泛函中心极限定理和精确渐近。我为FDA提出的方法绝对可以应用于基因表达研究，分析时间基因表达数据，并优化工具，以便随后筛选上游DNA序列，寻找可能解释具有相似时间行为的基因簇的共同序列基序。失效时间数据和纵向有序数据的研究成果将为生物医学和药学研究提供重要的方法。此外，极限定理的预期结果将为回归参数的自归一化型最小二乘估计的完全基于数据的渐近性质提供来源。

Functional data and failure time data frequently arise in many scientific fields such as biology, medical science, biomedical engineering, environmental engineering and bio-sciences. My proposed research program will mainly focus on developing a series of statistical methods and strategies to systematically analyze such data and then to explain the behaviors of the subjects from which the data are collected. Recently functional data analysis (FDA) has attracted the attention of many researchers and several approaches have been developed. The key to existing approaches is to represent high dimensional data by low dimensional variables, which can be analyzed using the methodology of multivariate random variables. However, if the dimensions of observations are ultra high, direct generalization of multivariate techniques to the realm of functional analysis is not in general feasible. Therefore I plan to focus on FDA from a different point of view and to develop some approaches to this research area. I will also develop the methodology to analyze the multivariate interval censored data. I will be considering the joint modeling for medical cost data with censoring mechanisms and the analysis for the longitudinal ordinal data from clinic trials. Limit theorems for self normalized sums of real valued and Banach space valued random variables have been my research interests. I will obtain results for the self-normalized sums such as law of iterated logarithm, functional central limit theorem, and precise asymptotics. My proposed methods for FDA can definitely be applied to the research of gene expression to analyze temporal gene expression data and to optimize tools for the consequent screening of upstream DNA sequences for common sequence motifs that might explain gene clusters showing similar time behaviors. The research results for the failure time data and longitudinal ordinal data will provide important methods for biomedical and pharmaceutical research. Furthermore, the expected results on limit theorems will provide a source for the completely data-based asymptotic properties of the bootstrapped Student t-statistic, self-normalized type least squares estimators of regression parameters.