  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.
