Statistical inference in time series analysis has been an important subject in various fields including climate science, economics, finance and industrial engineering among others. Numerous problems of research interest include statistical inference about unknown quantities, assessing structural stability and forecasting. These problems have been widely studied in the literature, but mainly for independent data, while in many applications involving time series data dependence is not unusual and in fact quite common. In this thesis, we incorporate serial dependence into the analysis by involving self-normalization in time series analysis. We start with the problem of testing whether there are change-points in a given time series. The method we propose does not require the number of change-points to be predefined, and thus is unsupervised. It does not require any tuning parameters and can be applied to a wide class to quantities of interest. The asymptotic distribution of the test statistic is studied and an approximation scheme is proposed to reduce testing procedure complexity. We then consider the problem of construction of confidence intervals, for which the conventional self-normalizer exhibits certain degrees of asymmetry when applied to quantities other than the mean. The method we propose provides a time-symmetric generalization to the conventional self-normalizer and leads to improved finite sample performance for quantities other than the mean.

Doctor of Philosophy / Department of Statistics / Weixin Yao / This dissertation contains two projects that are related to varying coefficient models. The traditional least squares based kernel estimates of the varying coefficient model will lose some efficiency when the error distribution is not normal. In the first project, we propose a novel adaptive estimation method that can adapt to different error distributions and provide an efficient EM algorithm to implement the proposed estimation. The asymptotic properties of the resulting estimator is established. Both simulation studies and real data examples are used to illustrate the finite sample performance of the new estimation procedure. The numerical results show that the gain of the adaptive procedure over the least squares estimation can be quite substantial for non-Gaussian errors. In the second project, we propose a unified inference for sparse and dense longitudinal data in time-varying coefficient models. The time-varying coefficient model is a special case of the varying coefficient model and is very useful in longitudinal/panel data analysis. A mixed-effects time-varying coefficient model is considered to account for the within subject correlation for longitudinal data. We show that when the kernel smoothing method is used to estimate the smooth functions in the time-varying coefficient model for sparse or dense longitudinal data, the asymptotic results of these two situations are essentially different. Therefore, a subjective choice between the sparse and dense cases may lead to wrong conclusions for statistical inference. In order to solve this problem, we establish a unified self-normalized central limit theorem, based on which a unified inference is proposed without deciding whether the data are sparse or dense. The effectiveness of the proposed unified inference is demonstrated through a simulation study and a real data application.

Utilizing multiple biomarkers in medical research is crucial for the diagnostic accuracy of detecting diseases. An optimal method for combining these biomarkers is essential to maximize the Area Under the Receiver Operating Characteristic (ROC) Curve (AUC). The optimality of the likelihood ratio has been proven but the challenges persist in estimating the likelihood ratio, primarily on the estimation of multivariate density functions. In this study, we propose a non-parametric approach for estimating multivariate density functions by utilizing Smoothing Spline density estimation to approximate the full likelihood function for both diseased and non-diseased groups, which compose the likelihood ratio. Simulation results demonstrate the efficiency of our method compared to other biomarker combination techniques under various settings for generated biomarker values. Additionally, we apply the proposed method to a real-world study aimed at detecting childhood autism spectrum disorder (ASD), showcasing its practical relevance and potential for future applications in medical research. Change point detection for functional time series has attracted considerable attention from researchers. Existing methods either rely on FPCA, which may perform poorly with complex data, or use bootstrap approaches in forms that fall short in effectively detecting diverse change functions. In our study, we propose a novel self-normalized test for functional time series implemented via a non-overlapping block bootstrap to circumvent reliance on FPCA. The SN factor ensures both monotonic power and adaptability for detecting diverse change functions on complex data. We also demonstrate our test's robustness in detecting changes in the autocovariance operator. Simulation studies confirm the superior performance of our test across various settings, and real-world applications further illustrate its practical utility. / Doctor of Philosophy / In medical research, it is crucial to accurately detect diseases and predict patient outcomes using multiple health indicators, also known as biomarkers. Combining these biomarkers effectively can significantly improve our ability to diagnose and treat various health conditions. However, finding the best way to combine these biomarkers has been a long-standing challenge. In this study, we propose a new, easy-to-understand method for combining multiple biomarkers using advanced estimation techniques. Our method takes into account various factors and provides a more accurate way to evaluate the combined information from different biomarkers. Through simulations, we demonstrated that our method performs better than other existing methods under a variety of scenarios. Furthermore, we applied our new method to a real-world study focusing on detecting childhood autism spectrum disorder (ASD), highlighting its practical value and potential for future applications in medical research. Detecting changes in patterns over time, especially shifts in averages, has become an important focus in data analysis. Existing methods often rely on techniques that may not perform well with more complex data or are limited in the types of changes they can detect. In this study, we introduce a new approach that improves the accuracy of detecting changes in complex data patterns. Our method is flexible and can identify changes in both the mean and variation of the data over time. Through simulations, we demonstrate that this approach is more accurate than current methods. Furthermore, we applied our method to real-world climate research data, illustrating its practical value.