Some theory about efficient dimension reduction regarding the interaction between two responses
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关于两个响应变量交互关系的有效降维理论
- 研究问题
- 如何刻画两个响应变量交互关系对应的有效降维空间，尤其是其边界、可估计性、稀疏性与空间唯一性？
- 方法与思路
- 作者分析双重逆回归子空间族的显式形式、上下界及一致且穷尽的估计，并将部分结果扩展到一般降维空间。
- 材料陈述的贡献
- 摘要称给出了子空间族的系统刻画及空间唯一性的可评估充要条件，并以模拟和真实数据展示因果推断应用。
- 阅读关注点
- 应核对预测变量条件、各子空间之间的关系、穷尽估计与唯一性的定义，以及因果应用额外需要的识别条件。
- 阅读边界
- 仅据所给 Crossref 摘要，无法判断所谓温和条件、定理范围或因果识别要求；应用展示本身不证明普遍适用。
原文摘要
Efficient dimension reduction regarding the interaction between two response variables, which facilitates statistical analysis in multiple important application scenarios, was initially discussed by Luo (J. R. Stat. Soc. Ser. B. Stat. Methodol. 84 (2022) 269–294). The efficient dimension reduction subspaces were introduced, and, under mild conditions on the predictor, they were equated with the family of dual inverse regression subspaces. Besides the general framework, however, limited theory has been proposed to uncover the mystery of these spaces. In this paper, we propose a thorough characterization of the family of dual inverse regression subspaces, including their uniform lower and upper bounds, their explicit forms, their consistent and exhaustive estimation, some interesting special cases, and certain subfamilies that have desired sparsity. In addition, we extend some of these results to provide more insights into the efficient dimension reduction subspaces under the general settings, including their uniform lower and upper bounds, and the sufficient and necessary conditions for the uniqueness of the space that are assessable in practice. These results largely complete the theoretical foundation for the new type of dimension reduction, and, as such, enhance its applicability in statistical problems such as missing data analysis and causal inference. The latter is illustrated by simulation studies and a real data example at the end.