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 → ∞ . However, the consistency as the noise variance σ² → 0 has not been studied. In this paper, we find that the MDL is inconsistent as σ² → 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 σ² → 0. Therefore, in a high signal-to-noise (SNR) scenario, the EEF provides a better criterion to use for model order selection.