Sufficient Dimension Folding for Regression Mean Function

In this article, we consider sufficient dimension folding for the regression mean function when predictors are matrix- or array-valued. We propose a new concept named central mean dimension folding subspace and its two local estimation methods: folded outer product of gradients estimation (folded-OPG) and folded minimum average variance estimation (folded-MAVE). We establish the asymptotic properties for folded-MAVE. A modified BIC criterion is used to determine the dimensions of the central mean dimension folding subspace. We evaluate the performances of the two local estimation methods by simulated examples and demonstrate the efficacy of folded-MAVE in finite samples. And in particular, we apply our methods to analyze a longitudinal study of primary biliary cirrhosis. Supplementary materials for this article are available online.

本文针对预测变量为矩阵或数组型的回归均值函数问题，探讨充分维度折叠方法。本文提出一种全新概念，命名为中心均值维度折叠子空间（central mean dimension folding subspace），并给出其两种局部估计方案：折叠梯度外积估计（folded-OPG）与折叠最小平均方差估计（folded-MAVE）。本文推导了折叠最小平均方差估计的渐近性质，采用修正贝叶斯信息准则确定中心均值维度折叠子空间的维度。通过仿真实验评估了两种局部估计方法的性能，并验证了折叠最小平均方差估计在有限样本下的有效性。特别地，本文将所提方法应用于原发性胆汁性肝硬化的纵向研究数据分析。本文的补充材料可在线获取。