AK-fold averaging cross-validation procedure

Cross-validation type of methods have been widely used to facilitate model estimation and variable selection. In this work, we suggest a new K-fold cross validation procedure to select a candidate 'optimal' model from each hold-out fold and average the K candidate 'optimal' models to obtain the ultimate model. Due to the averaging effect, the variance of the proposed estimates can be significantly reduced. This new procedure results in more stable and efficient parameter estimation than the classical K-fold cross validation procedure. In addition, we show the asymptotic equivalence between the proposed and classical cross validation procedures in the linear regression setting. We also demonstrate the broad applicability of the proposed procedure via two examples of parameter sparsity regularization and quantile smoothing splines modeling. We illustrate the promise of the proposed method through simulations and a real data example.