Likelihood theory, prediction, model selection: asymptotic connections.

Plug-in estimation and corresponding refinements involving penalisation have been considered in various areas of parametric statistical inference. One major example is adjustment of the profile likelihood for inference in the presence of nuisance parameters. Another important setting is prediction, where improved estimative predictive densities have been recently developed. A third related setting is model selection, where information criteria based on penalisation of maximised likelihood have been proposed starting from the pioneering contribution of Akaike. The seminal contributions in the last setting predate those introducing the former two classes of procedures, and pertinent portions of literature seem to have evolved quite independently. The aim of this paper is to establish some simple asymptotic connections among these classes of procedures. In particular, all the three kinds of penalisations involved can be viewed as bias corrections of plug-in estimates of theoretical target criteria which are shown to be very closely connected. As a by-product, we obtain adjusted profile likelihoods from optimal predictive densities. Links between adjusted procedures in likelihood theory and model selection procedures are also briefly enquired throuh some simulation studies.