Uniform Consistency of Generalized Cross-Validation for Ridge Regression in High-Dimensional Misspecified Linear Models

This study examines generalized cross-validation for the tuning parameter selection for ridge regression in high-dimensional misspecified linear models. The set of candidates for the tuning parameter includes not only positive values but also zero and negative values. We demonstrate that if the second moment of the specification error converges to zero, generalized cross-validation is still a uniformly consistent estimator of the out-of-sample prediction risk. This implies that generalized cross-validation selects the tuning parameter for which ridge regression asymptotically achieves the smallest prediction risk among the candidates if the degree of misspecification for the regression function is small. Our simulation studies show that ridge regression tuned by generalized cross-validation exhibits a prediction performance similar to that of optimally tuned ridge regression and outperforms the Lasso under correct and incorrect model specifications.

💡 Research Summary

This paper investigates the performance of generalized cross‑validation (GCV) for selecting the ridge‑regression tuning parameter in high‑dimensional settings where the true regression function is misspecified as linear. The authors allow the candidate set Λ to contain positive, zero, and even negative values of the regularization parameter, reflecting recent findings that optimal λ can be non‑positive when p>n. Under a proportional asymptotic regime (p/n→γ∈(0,∞) with γ≠1) and standard moment assumptions on the covariates and noise, they introduce a mild misspecification condition: the second moment of the specification error δ_i = f(x_i)−x_iᵀβ₀ satisfies E