High-dimensional Non-Gaussian Single Index Models via Thresholded Score Function Estimation

We consider estimating the parametric component​ ​of single index models in high dimensions.​ ​Compared with existing work,​ ​we do not require the covariate to be normally​ ​distributed. Utilizing Stein’s Lemma,​ ​we propose estimators based on the score​ ​function of the covariate. Moreover, to handle​ ​​score function and response variables​ ​that are heavy-tailed, our estimators are constructed​ ​via carefully thresholding their empirical​ ​counterparts. Under a bounded fourth​ ​moment condition, we establish optimal statistical​ ​rates of convergence for the proposed​ ​estimators. Extensive numerical experiments​ ​are provided to back up our theory.