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.