Variable selection in linear regression: Several approaches based on normalized maximum likelihood

The use of the normalized maximum likelihood (NML) for model selection in Gaussian linear regression poses troubles because the normalization coefficient is not finite. The most elegant solution has been proposed by Rissanen and consists in applying a particular constraint for the data space. In this paper, we demonstrate that the methodology can be generalized, and we discuss two particular cases, namely the rhomboidal and the ellipsoidal constraints. The new findings are used to derive four NML-based criteria. For three of them which have been already introduced in the previous literature, we provide a rigorous analysis. We also compare them against five state-of-the-art selection rules by conducting Monte Carlo simulations for families of models commonly used in signal processing. Additionally, for the eight criteria which are tested, we report results on their predictive capabilities for real life data sets.