THE RESTRICTED CONSISTENCY PROPERTY OF LEAVE-nv-OUT CROSS-VALIDATION FOR HIGH-DIMENSIONAL VARIABLE SELECTION

Cross-validation (CV) methods are popular for selecting the tuning parameter in high-dimensional variable selection problems. We show that a misalignment of the CV is one possible reason for its over-selection behavior. To fix this issue, we propose using a version of leave-n(v)-out CV (CV(n(v))) to select the optimal model from a restricted candidate model set for high-dimensional generalized linear models. By using the same candidate model sequence and a proper order for the construction sample size n(c) in each CV split, CV(n(v)) avoids potential problems when developing theoretical properties. CV(n(v)) is shown to exhibit the restricted model-selection consistency property under mild conditions. Extensive simulations and a real-data analysis support the theoretical results and demonstrate the performance of CV(n(v)) in terms of both model selection and prediction.