On the consistency of coordinate-independent sparse estimation with BIC

Chen et al. (2010) propose a unified method – coordinate-independent sparse estimation (CISE) – that is able to simultaneously achieve sparse sufficient dimension reduction and screen out irrelevant and redundant variables efficiently. However, its attractive features depend on appropriate choice of the tuning parameter. In this note, we re-examine the Bayesian information criterion (BIC) in sufficient dimension reduction and provide a heuristic derivation. Furthermore, the CISE with BIC is shown to be able to identify the true model consistently.