Robust parameter estimation of regression model under weakened moment assumptions

TL;DR

This paper extends parameter estimation techniques for regression models under weaker moment conditions, demonstrating phase transitions and proposing improved estimators for heavy-tailed data with theoretical and empirical validation.

Contribution

It introduces new estimators for regression models under weakened moment assumptions, including phase transition analysis and element-wise truncation for heavy-tailed data.

Findings

Robust estimators have slower convergence under weaker moments.

Phase transition phenomena are observed in the estimation process.

Proposed estimators perform well in simulations with heavy-tailed data.

Abstract

This paper provides some extended results on estimating parameter matrix of several 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 $1> \epsilon>0$, the robust estimator possesses a slower convergence rate compared with previous literature. 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.