We consider estimating the parametric components of semiparametric multi-index
models in high dimensions. To bypass the requirements of Gaussianity or elliptical
symmetry of covariates in existing methods, we propose to leverage a second-order
Stein’s method with score function-based corrections. We prove that our estimator
achieves a near-optimal statistical rate of convergence even when the score function
or the response variable is heavy-tailed. To establish the key concentration results,
we develop a data-driven truncation argument that may be of independent interest.
We supplement our theoretical findings with simulations.