Stable Estimation in Dimension Reduction

We introduce stable estimation procedures for several aspects of a sufficient dimension-reduction matrix. We first propose a stable method for estimating structural dimension, which only selects the correct directions in the central subspace with no false positive selection. We then provide a Grassmann manifold sparse estimate for the central subspace. By using subsampling, we develop an ensemble method to obtain a stable nonsparse estimate for the central subspace. This ensemble idea is also used to stabilize the choice of the number of slices in sliced inverse methods. Theoretical results are established, and the efficacy of the proposed stable methods is demonstrated by simulation studies and the analysis of Hitters’ salary data. Supplementary materials for this article are available online.

本文针对充分降维矩阵（sufficient dimension-reduction matrix）的多个层面，提出了一系列稳健估计方法。首先，我们提出一种用于估计结构维度的稳健方法，该方法仅能选取中心子空间（central subspace）中的正确方向，且不会产生假阳性选择。随后，我们针对中心子空间给出了基于格拉斯曼流形（Grassmann manifold）的稀疏估计量。通过采用子采样（subsampling）技术，我们构建了集成方法（ensemble method），以获取中心子空间的稳健非稀疏估计量。该集成思路同样可用于稳定切片逆方法（sliced inverse methods）中的切片数量选择环节。本文建立了相关理论结果，并通过模拟实验与Hitters薪资数据集（Hitters’ salary data）的实证分析，验证了所提稳健方法的有效性。本文的补充材料可在线获取。