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We propose a novel approach, called the weighted inverse regression ensemble method, for linear Fr\u00e9chet sufficient dimension reduction. The method is further generalized as a new operator defined on reproducing kernel Hilbert spaces for nonlinear Fr\u00e9chet sufficient dimension reduction. We provide theoretical guarantees for the new method via asymptotic analysis. Intensive simulation studies verify the performance of our proposals, and we apply our methods to analyse handwritten digit data and real-world affective face data to demonstrate its use in real applications.<\/jats:p>","DOI":"10.1093\/biomet\/asac012","type":"journal-article","created":{"date-parts":[[2022,2,15]],"date-time":"2022-02-15T08:38:20Z","timestamp":1644914300000},"page":"975-992","source":"Crossref","is-referenced-by-count":31,"title":["Fr\u00e9chet sufficient dimension reduction for random objects"],"prefix":"10.1093","volume":"109","author":[{"given":"Chao","family":"Ying","sequence":"first","affiliation":[{"name":"Key Laboratory of Advanced Theory and Application in Statistics and Data Science - MOE, School of Statistics , East China Normal University, Shanghai 200241, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhou","family":"Yu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Advanced Theory and Application in Statistics and Data Science - MOE, School of Statistics , East China Normal University, Shanghai 200241, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2022,2,17]]},"reference":[{"key":"2022112917594010400_B1","doi-asserted-by":"crossref","first-page":"1196","DOI":"10.1214\/aos\/1176345637","article-title":"Some asymptotic theory for the bootstrap","volume":"9","author":"Bickel,","year":"1981","journal-title":"Ann. 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