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Linear model selection by cross-validation. (English) Zbl 0773.62051

Summary: 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 $n\to \infty$.

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 ${n}_{v}/n\to 1$ as $n\to \infty$. This is a somewhat shocking discovery, because ${n}_{v}/n\to 1$ 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.

##### MSC:
 62J99 Linear statistical inference 65C99 Probabilistic methods, simulation and stochastic differential equations (numerical analysis)