Robust parameter estimation of regression model under weakened moment assumptions

This paper provides some extended results on estimating parameter matrix of some regression models when the covariate or response possesses weaker moment condition. We study the $M$-estimator of Fan et al. (Ann Stat 49(3):1239--1266, 2021) for matrix completion model with $(1+\epsilon)$-th moment noise. The corresponding phase transition phenomenon is observed. When $\epsilon \geq 1$, the robust estimator possesses the same convergence rate as previous literature. While $1> \epsilon>0$, the rate will be slower. For high dimensional multiple index coefficient model, we propose an improved estimator via applying the element-wise truncation method to handle heavy-tailed data with finite fourth moment. The extensive simulation study validates our theoretical results.Comment: 14 page