{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T17:51:11Z","timestamp":1791395471055,"version":"4.3.3"},"reference-count":32,"publisher":"Institute of Mathematical Statistics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2025,2,1]]},"abstract":"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\u2013294). 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.<\/jats:p>","DOI":"10.1214\/24-aos2463","type":"journal-article","created":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T01:35:51Z","timestamp":1739496951000},"page":"245-267","source":"Crossref","is-referenced-by-count":0,"title":["Some theory about efficient dimension reduction regarding the interaction between two responses"],"prefix":"10.1214","volume":"53","author":[{"given":"Wei","family":"Luo","sequence":"first","affiliation":[{"name":"Center for Data Science, Zhejiang University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"5","doi-asserted-by":"publisher","unstructured":"DE LUNA, X.<\/small>, WAERNBAUM, I.<\/small> and RICHARDSON, T. S.<\/small> (2011). Covariate selection for the nonparametric estimation of an average treatment effect. 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