Robust parameter estimation of regression model under weakened moment assumptions

Abstract

This paper provides some extended results on estimating the parameter matrix of high-dimensional regression model when the covariate or response possess weaker moment condition. We investigate the M -estimator of Fan et al. ( AnnStat 49(3):1239-1266, 2021) for matrix completion model with (1 + ǫ)-th moments. The corresponding phase transition phenomenon is observed. When ǫ ≥ 1, the robust estimator possesses the same convergence rate as previous literature. While 1 > ǫ > 0, the rate will be slower. For high dimensional multiple index coefficient model, we also apply the element-wise truncation method to construct a robust estimator which handle missing and heavy-tailed data with finite fourth moment.