Linear Model Selection by Cross-validation

Abstract

We consider the problem of selecting a model having the best predictive ability among a class of linear models. The popular leave-one-out cross-validation method, which is asymptotically equivalent to many other model selection methods such as the Akaike information criterion (AIC), the C _p , and the bootstrap, is asymptotically inconsistent in the sense that the probability of selecting the model with the best predictive ability does not converge to 1 as the total number of observations n → ∞. We show that the inconsistency of the leave-one-out cross-validation can be rectified by using a leave-n _v -out cross-validation with n _v , the number of observations reserved for validation, satisfying n _v /n → 1 as n → ∞. This is a somewhat shocking discovery, because n_v/n → 1 is totally opposite to the popular leave-one-out recipe in cross-validation. Motivations, justifications, and discussions of some practical aspects of the use of the leave-n _v -out cross-validation method are provided, and results from a simulation study are presented.

Key Words:

• Balanced incomplete
• Consistency
• Data splitting
• Model assessment
• Monte Carlo
• Prediction