Model-free Envelope Dimension Selection

TL;DR

This paper introduces two model-free methods, FG and 1D selections, for consistently estimating the envelope dimension in envelope models, enhancing theoretical justification and computational stability.

Contribution

It proposes two unified, model-free approaches for envelope dimension selection applicable to any envelope models, with theoretical guarantees and practical advantages.

Findings

Both methods correctly identify the envelope dimension with high probability as sample size increases.

FG selection generalizes BIC approaches with theoretical justification under weak conditions.

1D selection offers computational stability and efficiency in finite samples.

Abstract

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