Asymptotic optimality of a cross-validatory predictive approach to linear model selection

TL;DR

This paper proves that a cross-validatory predictive model selection method for linear models is asymptotically optimal, performing as well as an oracle that knows the true regression function, under squared error loss.

Contribution

It establishes the asymptotic optimality of a cross-validatory predictive criterion for linear model selection, extending understanding of its theoretical performance.

Findings

The method achieves asymptotic predictive optimality.

It performs as well as an oracle with knowledge of the true model.

The results are valid under squared error prediction loss.

Abstract

In this article we study the asymptotic predictive optimality of a model selection criterion based on the cross-validatory predictive density, already available in the literature. For a dependent variable and associated explanatory variables, we consider a class of linear models as approximations to the true regression function. One selects a model among these using the criterion under study and predicts a future replicate of the dependent variable by an optimal predictor under the chosen model. We show that for squared error prediction loss, this scheme of prediction performs asymptotically as well as an oracle, where the oracle here refers to a model selection rule which minimizes this loss if the true regression were known.