Functional Outlier Detection for Density-Valued Data with Application to Robustify Distribution-to-Distribution Regression

Distributional data analysis, concerned with the statistical analysis of data objects consisting of random probability distributions in the framework of functional data analysis (FDA), has received considerable interest in recent years and is increasingly applied in various fields including engineering. Outlier detection and robustness are of great practical interest; however, these aspects remain unexplored for distributional data. To this end, this study focuses on density-valued outlier detection and its application in robust distributional regression. Specifically, we propose a transformation-based approach for single-dataset outlying density detection with an emphasis on converting the less detectable shape outliers to easily detectable magnitude outliers. We also propose a distributional regression-based approach for detecting the abnormal associations of the density-valued two-tuples associated with two datasets. Then, the proposed outlier detection methods are applied to robustify a distribution-to-distribution regression method used in engineering, and we develop a robust estimator for the regression operator by downweighting the detected outliers. The proposed methods are validated and evaluated via extensive simulation studies. The relevant results reveal the superiority of our method over other competitors in distributional outlier detection. A case study in structural health monitoring demonstrates the great potential of our proposal in engineering applications. Supplementary materials for this article are available online.