Robust Inference for High‐Dimensional Single Index Models

Abstract

We propose a robust inference method for high‐dimensional single index models with an unknown link function and elliptically symmetrically distributed covariates, focusing on signal recovery and inference. The proposed method is built on the Huber loss and the estimation of the unknown link function is avoided. The ℓ1$$ {\ell}_1 $$ and ℓ2$$ {\ell}_2 $$ consistency of a Lasso estimator up to a multiplicative scalar is established. When the covariance matrix of the predictors satisfies the irrepresentable condition, our method is shown to recover the signed support of the true parameter under mild conditions. Based on a debiased Lasso estimator, we study component‐wise and group inference for the high‐dimensional index parameter. The finite‐sample performance of our method is evaluated through extensive simulation studies. An application to a riboflavin production dataset is provided to illustrate the proposed method.