AI 中文总结
针对组间回归关系存在差异的多元线性回归场景，提出分组预测器包络模型，其渐近效率优于分组拟合的方法，模拟研究验证了该理论优势。
AI 中文摘要
包络方法通过分离包含目标参数所有相关信息的低维结构，提升多元分析中的估计效率。在具有随机预测变量的多元线性回归中，预测器包络模型通过剔除预测变量中与回归无关的变异实现这一目标。在许多应用场景中，观测值会自然划分为若干组，如处理组、区域或人口统计层，且组间回归关系可能存在差异。基于该设定，我们提出了一种用于多元线性回归的分组预测器包络模型。该模型假设各组特定的回归系数矩阵通过一个公共预测器包络子空间表示，同时允许存在组特定的回归效应和组特定的误差协方差矩阵。我们推导了估计公共预测器包络的目标函数，得到了对应的回归估计量，并建立了渐近正态性及显式的渐近方差公式。此外，我们证明了所提估计量在渐近意义上比将预测器包络模型分别拟合至每个组得到的估计量更有效，模拟研究也验证了该理论优势。
英文摘要
Envelope methods improve estimation efficiency in multivariate analysis by isolating low-dimensional structures that contain all the information material to the parameter of interest. In multivariate linear regression with random predictors, predictor envelope models achieve this goal by removing variation in the predictors that is immaterial to the regression. In many applications, observations are naturally divided into several groups, such as treatment groups, regions, or demographic strata, and the regression relationship may differ between groups. Motivated by this setting, we propose a groupwise predictor envelope model for multivariate linear regression. The proposed model assumes that the group-specific regression coefficient matrices are represented through a common predictor envelope subspace while allowing group-specific regression effects and group-specific error covariance matrices. We derive an objective function for estimating the common predictor envelope, obtain the corresponding regression estimators, and establish asymptotic normality together with an explicit asymptotic variance formula. Moreover, we show that the proposed estimator is asymptotically more efficient than the estimator obtained by fitting predictor envelope models separately to each group. This theoretical advantage over the existing work is also demonstrated through simulation studies.
