Date of Award

12-16-2019

Degree Type

Closed Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Yichuan Zhao, PhD

Second Advisor

Xin Qi, PhD

Third Advisor

Ruiyan Luo, PhD

Fourth Advisor

Jing Zhang, PhD

Fifth Advisor

Jun Kong, PhD

Abstract

There have been several advances in statistical inference and computation procedures, whereas advancement in technology facilitates extensive computations. We consider two efficient approaches, i.e., the functional data analysis technique and the empirical likelihood inference procedures. These techniques require extensive computational procedures. For the functional data analysis part, we consider a functional autoregressive (FAR) model with general order. There exist several literatures on classic time series, but a limited number of works have been done on functional time series. We propose a signal compression procedure to fit the FAR model and to forecast future observations. To determine the optimal tuning parameters and optimal order of FAR model, we propose a window-shifting cross-validation procedure. We compare the model to recently proposed method on both simulated data and real data, which illustrate good predictive performance of our method.

For the diagnostic medicine portion of this dissertation, we propose empirical likelihood (EL) inference procedures for two motivating statistical measures in biomedical research, i.e., the two-way partial AUC (tpAUC) and the sensitivity to the early disease stage. The EL procedure does not require underlying distributional assumption and is very suitable for drawing the inference about parameters. The area under the curve (AUC) is a summary measure for the receiver operating characteristic (ROC) curve. There have been no good confidence intervals proposed for the tpAUC. Motivated by this lacking, we propose EL confidence interval for the tpAUC and the difference between two tpAUCs. We also propose EL confidence interval for the sensitivity to the early disease stage and for the difference between two sensitivities to early disease stages. The early disease stage can play a vital role for the therapeutic intervention and prevention potentiality. Better inference procedure for the sensitivity can ensure the identification of better performing biomarkers. Our extensive simulation studies suggested good performance of the proposed procedures compared to the existing methods. Finally, real data sets are analyzed for illustration of the proposed methods.

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