Abstract Sufficient dimension reduction that replaces original predictors with their low- dimensional linear combinations without loss of information is a critical tool in modern statistics and has gained considerable research momentum in the past decades since the two pioneers sliced inverse regression and principal Hessian directions. The classical sufficient dimension reduction methods do not handle sparse case well since the estimated linear reductions involve all of the original predictors. Sparse sufficient dimension reduction methods rely on sparsity assumption which may not be true in practice. Motivated by the least squares formulation of the classical sliced inverse regression and principal Hessian directions, several model averaging assisted sufficient dimension reduction methods are proposed. They are applicable to both dense and sparse cases even with weak signals since model averaging adaptively assigns weights to different candidate models. Based on the model averaging assisted sufficient dimension reduction methods, how to estimate the structural dimension is further studied. Theoretical justifications are given and empirical results show that the proposed methods compare favorably with the classical sufficient dimension reduction methods and popular sparse sufficient dimension reduction methods.

中文翻译：

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模型平均辅助充分降维

摘要 在不丢失信息的情况下用低维线性组合代替原始预测变量的充分降维是现代统计学中的关键工具，自从两位先驱将逆回归和主要 Hessian 方向切片以来，在过去的几十年中获得了相当大的研究动力。经典的充分降维方法不能很好地处理稀疏情况，因为估计的线性减少涉及所有原始预测变量。稀疏充分降维方法依赖于稀疏假设，这在实践中可能不正确。受经典切片逆回归和主要 Hessian 方向的最小二乘公式的启发，提出了几种模型平均辅助的充分降维方法。它们适用于密集和稀疏情况，即使信号较弱，因为模型平均会自适应地为不同的候选模型分配权重。基于模型平均辅助的充分降维方法，进一步研究了结构维数的估计方法。给出了理论依据和经验结果表明，所提出的方法与经典的充分降维方法和流行的稀疏充分降维方法相比具有优势。