Informational complexity criteria for regression models

Abstract

This paper pursues three objectives in the context of multiple regression models: • • To give a rationale for model selection criteria which combine a badness-of-fit term (such as minus twice the maximum log likelihood) with a measure of complexity of model. We show that the ICOMP criterion introduced by Bozdogan can be seen as an approximation to the sum of two Kullback-Leibler distances, and that a criterion related to ICOMP arises as an approximation to the posterior expectation of a certain utility. • • To investigate the asymptotic consistency properties of the class of ICOMP criteria first in the case when one of the models considered is the true model and to introduce and establish a consistency property for the case when none of the models is the true model. In the first case, we find that asymptotic consistency holds under some assumptions; in this respect, some ICOMP criteria resemble Akaike's AIC, while other ICOMP criteria resemble Schwarz's BIC criterion. In the second case, in the context of regression models where at least one independent variable is missing in each model, we find that ICOMP, as well as AIC and BIC are all asymptotically consistent. • • To investigate the finite sample behavior of ICOMP criteria by means of a simulation study where none of the models considered is the true model. We find that the ICOMP criteria tend to agree with decisions based on minimizing the Kullback-Leibler distance between the true model and each estimated model more often than AIC or BIC. This conclusion also holds when the true model is one of the models considered.