High‐Dimensional Inference for Single‐Index Models With Latent Factors

ABSTRACT Models with latent factors have recently attracted considerable attention. However, most existing studies focus on linear regression models and therefore fail to capture potential nonlinear structures. To address this limitation, we consider the factor augmented single‐index model. We first examine whether the inclusion of the augmented component is necessary by introducing a score‐type test statistic. Unlike existing test statistics, the proposed one does not require estimating high‐dimensional regression coefficients or precision matrices, making it computationally simpler and more stable. To determine the critical value, we employ a Gaussian multiplier bootstrap, whose theoretical validity is established under mild regularity conditions. We further investigate the penalized estimation of the regression model. With estimated latent factors, we establish the error bounds of the estimators. In addition, we construct confidence intervals for individual coefficients based on a debiased estimator. Importantly, our method does not impose moment conditions on the error distribution, allowing it to perform well even when the errors are heavy‐tailed or contaminated by outliers. Comprehensive simulation studies and an application to a gene expression dataset demonstrate the effectiveness and robustness of the proposed procedure.