Advances in robust multivariate statistics : methods for component analysis, causal inference, and dependence modeling

Abstract

This dissertation advances the field of robust multivariate statistics by developing new methods that are well-suited for data containing outliers and heavy tails. By introducing targeted robust procedures, including specialized estimators and tailored data transformations, we adapt four existing statistical methods and demonstrate their effectiveness through simulations and real-world applications.​​ First, we propose a computationally efficient robust principal component analysis method (PCA) based on decomposing the generalized spatial sign covariance matrix. This approach overcomes the computational bottlenecks and sample-size limitations of existing robust PCA methods while maintaining flexibility through various radial functions. Second, in the domain of causal inference, we introduce TSLiNGAM. By replacing the non-robust ordinary least squares regression in the original LiNGAM algorithm with Theil-Sen regression, this method achieves superior causal discovery on heavy-tailed and skewed data. Third, addressing the need for a highly robust independence measure, we introduce the biloop data transformation to robustify distance correlation. The resulting biloop distance correlation achieves a bounded, redescending influence function and a non-zero breakdown point. Fourth, we leverage the biloop distance correlation to develop a new approach for independent component analysis (ICA). By applying a novel transform called bowl to the dCovICA method, we effectively separate multivariate sources in the presence of outliers. Finally, to bridge the gap between robust statistics, traditionally more established in R, and Python-based data analysis, we present RobPy, a comprehensive Python library for robust regression, PCA, covariance estimation and cellwise outlier detection.