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In many conventional scientific investigations with high or ultra-highdimensional feature spaces, the relevant features, though sparse, are large innumber compared with classical statistical problems, and the magnitude of theireffects tapers off. It is reasonable to model the number of relevant featuresas a diverging sequence when sample size increases. In this article, weinvestigate the properties of the extended Bayes information criterion (EBIC)(Chen and Chen, 2008) for feature selection in linear regression models withdiverging number of relevant features in high or ultra-high dimensional featurespaces. The selection consistency of the EBIC in this situation is established.The application of EBIC to feature selection is considered in a two-stagefeature selection procedure. Simulation studies are conducted to demonstratethe performance of the EBIC together with the two-stage feature selectionprocedure in finite sample cases.

Extended BIC for linear regression models with diverging number of relevant features and high or ultra-high feature spaces