Overlapping group-LASSO for quantile regression: methodology, computation, and functional data applications

In this thesis we combined linear quantile regression with the Jenatton et. al. (2011) overlap group LASSO penalty. An exact and an approximate problem formulations were considered. In order to solve the resulting optimization problem, conjoint ADMM-MM schemes were employed. In the approximated formulation, an explicit upper bound formula for the maximum penalty parameter beyond which all group penalized coefficients are shrunk to zero was derived. We evaluated the empirical properties of the proposed methods through a series of simulations covering linear regression, B-spline smoothing, and scalar-on-function regression scenarios, the latter case presents also a real data application. Overall, the proposed quantile methods outperformed their squared-loss counterparts under error distributions violating standard least-squares assumptions, particularly in all heavy-tailed scenarios and in the majority of (normal) heteroscedastic ones. Crucially, this advantage manifested in estimation and prediction metrics, whereas false positive rate was generally greater than in the squared loss methods. While there is considerable room to refine and extend these frameworks, this work paves the way for further methodological innovations in quantile-based regression.