Functional outlier detection for density-valued data with application to robustify distribution to distribution regression

Distributional data analysis, concerned with statistical analysis and modeling for data objects consisting of random probability density functions (PDFs) in the framework of functional data analysis (FDA), has received considerable interest in recent years. However, many important aspects remain unexplored, such as outlier detection and robustness. Existing functional outlier detection methods are mainly used for ordinary functional data and usually perform poorly when applied to PDFs. To fill this gap, this study focuses on PDF-valued outlier detection, as well as its application in robust distributional regression. Similar to ordinary functional data, detecting the shape outlier masked by the "curve net" formed by the bulk of the PDFs is the major challenge in PDF-outlier detection. To this end, we propose a tree-structured transformation system for feature extraction as well as converting the shape outliers to easily detectable magnitude outliers, relevant outlier detectors are designed for the specific transformed data. A multiple detection strategy is also proposed to account for detection uncertainties and to combine different detectors to form a more reliable detection tool. Moreover, we propose a distributional-regression-based approach for detecting the abnormal associations of PDF-valued two-tuples. As a specific application, the proposed outlier detection methods are applied to robustify a distribution-to-distribution regression method, and we develop a robust estimator for the regression operator by downweighting the detected outliers. The proposed methods are validated and evaluated by extensive simulation studies or real data applications. Relevant comparative studies demonstrate the superiority of the developed outlier detection method with other competitors in distributional outlier detection.