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Distributional data analysis, concerned with statistical analysis andmodeling for data objects consisting of random probability density functions(PDFs) in the framework of functional data analysis (FDA), has receivedconsiderable interest in recent years. However, many important aspects remainunexplored, such as outlier detection and robustness. Existing functionaloutlier detection methods are mainly used for ordinary functional data andusually perform poorly when applied to PDFs. To fill this gap, this studyfocuses on PDF-valued outlier detection, as well as its application in robustdistributional regression. Similar to ordinary functional data, detecting theshape outlier masked by the "curve net" formed by the bulk of the PDFs is themajor challenge in PDF-outlier detection. To this end, we propose atree-structured transformation system for feature extraction as well asconverting the shape outliers to easily detectable magnitude outliers, relevantoutlier detectors are designed for the specific transformed data. A multipledetection strategy is also proposed to account for detection uncertainties andto combine different detectors to form a more reliable detection tool.Moreover, we propose a distributional-regression-based approach for detectingthe abnormal associations of PDF-valued two-tuples. As a specific application,the proposed outlier detection methods are applied to robustify adistribution-to-distribution regression method, and we develop a robustestimator for the regression operator by downweighting the detected outliers.The proposed methods are validated and evaluated by extensive simulationstudies or real data applications. Relevant comparative studies demonstrate thesuperiority of the developed outlier detection method with other competitors indistributional outlier detection.

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