High SNR Consistent Linear Model Order Selection and Subset Selection

High SNR consistency of model order selection criteria in linear regression models has attracted a lot of attention in the signal processing community recently. It is now known that Exponentially Embedded Family, versions of Minimum Description Length etc. are high SNR consistent. However, a general framework for high SNR consistency in linear regression is still missing. This paper fills this gap by deriving necessary and sufficient conditions (NSCs) for model order selection criteria in both known and unknown noise variance situations to be high SNR consistent. Many popular model order selection criteria are proved to be high SNR consistent using these NSCs. We also provide a convergence rate analysis to discriminate between various high SNR consistent model order selection criteria. A direct application of model order selection techniques to the very important subset selection problem is computationally infeasible. This paper establish the high SNR consistency of an existing t-statistics based index ordering scheme that allows the conversion of a subset selection problem into a model order selection problem. Combining this index ordering with high SNR consistent model order selection criteria leads to a novel subset selection procedure (SSP) that is numerically shown to outperform existing high SNR consistent SSPs.