Inconsistency of the MDL: On the Performance of Model Order Selection Criteria With Increasing Signal-to-Noise Ratio

In the problem of model order selection, it is well known that the widely used minimum description length (MDL) criterion is consistent as the sample size N -> infinity. However, the consistency as the noise variance sigma(2) -> 0 has not been studied. In this paper, we find that the MDL is inconsistent as sigma(2) -> 0. The result shows that the MDL has a tendency to overestimate the model order. We also prove that another criterion, the exponentially embedded family (EEF), is consistent as sigma(2) -> 0. Therefore, in a high signal-to-noise (SNR) scenario, the EEF provides a better criterion to use for model order selection.