High-dimensional Inference Via Hybrid Orthogonalization

The past decade has witnessed a surge of endeavors in statistical inference for high-dimensional sparse regression, particularly via de-biasing or relaxed orthogonalization.Nevertheless, these techniques typically require a more stringent sparsity condition than is needed for estimation consistency, which seriously limits their practical applicability.To alleviate such a constraint, we propose to exploit the strong predictors to residualize the design matrix before performing debiasing-based inference on the parameters of interest.This leads to a hybrid orthogonalization technique (HOT) that performs strict orthogonalization against the strong predictors but relaxed orthogonalization against the others.Under an approximately sparse model with a mixture of strong and weak signals, we establish the asymptotic normality of the HOT estimator while accommodating as many strong signals as consistent estimation allows.The efficacy of the proposed method is also demonstrated through numerical studies.