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For mean estimation with respect to squared Euclidean error loss, we show that the minimax quantiles decompose as a sum of the corresponding minimax quantiles under a heterogeneous, MCAR assumption and a robust error term, depending on \u03f5, that reflects the additional error incurred by departure from MCAR. We next introduce natural classes of realisable \u03f5-contamination models, where an MCAR version of a base distribution P is contaminated by an arbitrary missing not at random (MNAR) version of P. These classes are rich enough to capture various notions of biased sampling and sensitivity conditions, yet we show that they enjoy improved minimax performance relative to our earlier arbitrary contamination classes for both parametric and nonparametric classes of base distributions. For instance, with a univariate Gaussian base distribution, consistent mean estimation over realisable \u03f5-contamination classes is possible even when \u03f5 and the proportion of missingness converge (slowly) to 1. We extend our results to the setting of departures from missing at random (MAR) in normal linear regression with a realisable missing response, and also demonstrate that our methods can be made adaptive to the case of unknown \u03f5.<\/jats:p>","DOI":"10.1214\/26-aos2640","type":"journal-article","created":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T08:19:05Z","timestamp":1787041145000},"page":"2055-2082","source":"Crossref","is-referenced-by-count":0,"title":["Estimation beyond missing (completely) at random"],"prefix":"10.1214","volume":"54","author":[{"given":"Tianyi","family":"Ma","sequence":"first","affiliation":[{"name":"Statistical Laboratory, University of Cambridge"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kabir A.","family":"Verchand","sequence":"additional","affiliation":[{"name":"Department of Data Sciences and Operations, University of Southern California"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas B.","family":"Berrett","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Warwick"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tengyao","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Statistics, London School of Economics and Political Science"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Richard J.","family":"Samworth","sequence":"additional","affiliation":[{"name":"Statistical Laboratory, University of Cambridge"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"9","doi-asserted-by":"publisher","unstructured":"Cai, T. 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