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In contrast to prior research focusing on parameter inference, this work explores the complexities of semi-supervised distribution estimation, particularly the uniformity problem inherent in functional processes. To tackle this issue, we introduce a versatile framework designed to extract valuable information from unlabelled data by approximating a conditional distribution on covariates. The proposed estimator is derived using K-fold cross-fitting, and exhibits both consistency and asymptotic Gaussian process properties. Under mild conditions, the proposed estimator outperforms the empirical cumulative distribution function in terms of asymptotic efficiency. Several applications of the methodology are given, including parameter inference and goodness-of-fit tests.<\/jats:p>","DOI":"10.1093\/biomet\/asae056","type":"journal-article","created":{"date-parts":[[2024,10,30]],"date-time":"2024-10-30T17:30:04Z","timestamp":1730309404000},"source":"Crossref","is-referenced-by-count":3,"title":["Semi-supervised distribution learning"],"prefix":"10.1093","volume":"112","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-4214-7973","authenticated-orcid":false,"given":"Mengtao","family":"Wen","sequence":"first","affiliation":[{"name":"School of Statistics and Data Science, Nankai University , 94 Weijin Road, Tianjin 300071, \u00a0 mtwen97@gmail.com \u00a0 yxjia@mail.nankai.edu.cn","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yinxu","family":"Jia","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Nankai University , 94 Weijin Road, Tianjin 300071, \u00a0 mtwen97@gmail.com \u00a0 yxjia@mail.nankai.edu.cn","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haojie","family":"Ren","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Shanghai Jiao Tong University , 800 Dongchuan Road , Shanghai 200240,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhaojun","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Nankai University , 94 Weijin Road , Tianjin 300071, \u00a0 zjwangnk@126.com \u00a0 nk.chlzou@gmail.com","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Changliang","family":"Zou","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Nankai University , 94 Weijin Road , Tianjin 300071, \u00a0 zjwangnk@126.com \u00a0 nk.chlzou@gmail.com","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2024,10,30]]},"reference":[{"key":"2025012017220515500_asae056-B1","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1126\/science.adi6000","article-title":"Prediction-powered inference","volume":"382","author":"Angelopoulos","year":"2023","journal-title":"Science"},{"key":"2025012017220515500_asae056-B2","first-page":"1","article-title":"Distributional random forests: heterogeneity adjustment and multivariate distributional regression","volume":"23","author":"\u0106evid","year":"2022","journal-title":"J. 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