Semiparametrically Efficient Method for Enveloped Central Space

The estimation of the central space is at the core of the sufficient dimension reduction (SDR) literature. However, it is well known that the finite-sample estimation suffers from collinearity among predictors. Cook, Helland, and Su proposed the predictor envelope method under linear models that can alleviate the problem by targeting a bigger space—which not only envelopes the central information, but also partitions the predictors by finding an uncorrelated set of material and immaterial predictors. One limitation of the predictor envelope is that it has strong distributional and modeling assumptions and therefore, it cannot be readily used in semiparametric settings where SDR usually nests. In this article, we generalize the envelope model by defining the enveloped central space and propose a semiparametric method to estimate it. We derive the entire class of regular and asymptotically linear (RAL) estimators as well as the locally and globally semiparametrically efficient estimators for the enveloped central space. Based on the connection between predictor envelope and partial least square (PLS), our methods can also be used to calculate the PLS space beyond linearity. In the simulations, our methods are shown to be both robust and accurate for estimating the enveloped central space under different settings. Moreover, the downstream analysis using state-of-the-art methods such as machine learning (ML) methods has the potential to achieve much better predictions. We further illustrate our methods in a heart failure study. Supplementary materials for this article are available online.