Statistical Methods for Functional Data and Failure Time Data

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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.