Asymptotically minimax regret procedures in regression model selection and the magnitude of the dimension penalty

Abstract

This paper addresses the topic of model selection in regression.We emphasize the case of two models, testing which model provides a better prediction based on $n$ observations. Within a family of selection rules, based on maximizing a penalized log-likelihood under a normal model, we search for asymptotically minimax rules over a class $\mathscr{G}$ of possible joint distributions of the explanatory and response variables. For the class $\mathscr{G}$ of multivariate normal joint distributions it is shown that asymptotically minimax selection rules are close to the AIC selection rule when the models’ dimension difference is large. It is further proved that under fairly mild assumptions on $\mathscr{G}$ any asymptotically minimax sequence of procedures satisfies the condition that the difference in their dimension penalties is bounded as the number of observations approaches infinity. The results are then extended to the case of more than two competing models.