Analysis And Modeling Of The Special Functional Data And Survival Data

Due to the advances in modern science and technology that enhance capability of data collection,storage and processing,complex data,including functional data and survival data,are commonly encountered in statistical area.The functional data can be seen as a function of time,and is often observed at finite time points.For survival data,the time that an event of interest occurs is incompletely recorded because of censoring.This dissertation,studying special functional data and survival data,is consist of four parts.Firstly,the dissertation studies the estimation of mean and covariance functions for functional snippet data.In functional data analysis,it is often assume that there are enough data in the domain of interest to estimate both the mean and covariance functions.However,the observations from a subject for functional snippets are available only in an interval of length strictly(and often much)shorter than the length of the whole interval of interest.For such a sampling plan,no data is available for direct estimation of the offdiagonal region of the covariance function.the dissertation tackles this challenge via a global basis representation of the covariance function.The proposed approach allows one to consistently estimate an infinite-rank covariance function from functional snippets.the dissertation also establishes the large sample properties of the proposed estimators.Secondly,the dissertation proposes a varying-coefficient additive model to fit the data with sparsely observed functional response and functional(or vector)covariates.The model generalizes both the varying-coefficient model and the additive model,and retains their merits as an effective dimension reduction model that is flexible yet easily interpretable.Consistency and rate of convergence are developed for the proposed estimators of the unknown functions.A algorithm is developed that overcomes the computational difficulty caused by the non-convexity of the objective function.This dissertation as well illustrate the approach through a simulation study and a real data application.Thirdly,this dissertation considers the application of functional data in survival analysis.The case-cohort design is generally used as a cost-efficient method associated with expensive covariate gathering in a large epidemiological cohort study.Methodologies for the case-cohort study are mainly developed for the vector covariates despite the recent explosion of functional covariates.With both functional and vector covariates,the dissertation studies the functional Cox model for the case-cohort study.the dissertation establishes the asymptotic properties of the maximum penalized pseudolikelihood estimator under the framework of reproducing kernel Hilbert space,and show that the vector estimator is consistent and has an asymptotic normal distribution.In addition,the dissertation developes the convergence rate of the functional coefficient estimator.The simulation studies and application in a Mediterranean fruit fly dat are carried out to demonstrate the performances of the proposed method.Finally,the dissertation applies deep learning to survival analysis.While deep neural networks excel in traditional supervised learning,it remains unclear how to best utilize these models in survival analysis.Rather than estimating the survival function targeted by most existing methods,dissertation introduces a Deep Accelerated Failure Time(DeepAFT)model that directly predicts the event time of each subject.the dissertation additionally proposes a Deep Extended Hazard(DeepEH)model to provide a more flexible and general framework for deep survival analysis.The proposed methods are proven to be asymptotically consistent,and empirically outperform existing statistical and deep learning approaches to survival analysis.