Model-free Envelope Dimension Selection

摘要

An envelope is a targeted dimension reduction subspace for simultaneouslyachieving dimension reduction and improving parameter estimation efficiency.While many envelope methods have been proposed in recent years, all envelopemethods hinge on the knowledge of a key hyperparameter, the structuraldimension of the envelope. How to estimate the envelope dimension consistentlyis of substantial interest from both theoretical and practical aspects.Moreover, very recent advances in the literature have generalized envelope as amodel-free method, which makes selecting the envelope dimension even morechallenging. Likelihood-based approaches such as information criteria andlikelihood-ratio tests either cannot be directly applied or have no theoreticaljustification. To address this critical issue of dimension selection, wepropose two unified approaches -- called FG and 1D selections -- fordetermining the envelope dimension that can be applied to any envelope modelsand methods. The two model-free selection approaches are based on the twodifferent envelope optimization procedures: the full Grassmannian (FG)optimization and the 1D algorithm (Cook and Zhang, 2016), and are shown to becapable of correctly identifying the structural dimension with a probabilitytending to 1 under mild moment conditions as the sample size increases. Whilethe FG selection unifies and generalizes the BIC and modified BIC approachesthat existing in the literature, and hence provides the theoreticaljustification of them under weak moment condition and model-free context, the1D selection is computationally more stable and efficient in finite sample.Extensive simulations and a real data analysis demonstrate the superbperformance of our proposals.