UC San Diego

High-dimensional data can often display heterogeneity due to heteroscedastic variance or inhomogeneous covariate effects. Penalized quantile and expectile regression methods offer useful tools to detect heteroscedasticity in high-dimensional data. However, the former is computationally challenging due to the non-smooth nature of the check loss, and the latter is sensitive to heavy-tailed error distributions. In this thesis, we propose and study (penalized) robust expectile regression (retire) with random designs and heavy-tailed noises. Theoretically, we establish the statistical properties of the retire estimator under two regimes: (i) low-dimensional regime in which d