Extension of the Schwarz Information Criterion for models sharing parameter boundaries

The classic Schwarz Information Criterion, originally derived as an approximation to Bayes posterior probability, is widely used as a standalone likelihood-based measure of model fit. However, selection consistency is compromised when model sets partially include their parameter boundaries and when these in turn are partially shared by different models. This happens, for example, where sets represent mixed weak and strict inequality restrictions on parameters. To enable consistent selection of such models, a generic extension of the Schwarz criterion is required but does not appear to be available in the literature to date. In this paper, we define the boundary extended Schwarz criterion for a model to be the maximum of Schwarz-type criteria applied to the model parameter space and a systematically-generated list of boundary subsets. This entails new concepts of boundary interaction level and model dimension. A self-contained theory is presented along with examples and simulation.