Likelihood Ratio Combination of Multiple Biomarkers and Change Point Detection in Functional Time Series

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.