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By leveraging a shadow variable, we establish a necessary and sufficient condition for identification of the mean functional even if the full data distribution is not identified. We further characterize a necessary condition for n-estimability of the mean functional. This condition naturally strengthens the identifying condition, and it requires the existence of a function as a solution to a representer equation that connects the shadow variable to the mean functional. Solutions to the representer equation may not be unique, which presents substantial challenges for non-parametric estimation, and standard theories for non-parametric sieve estimators are not applicable here. We construct a consistent estimator of the solution set and then adapt the theory of extremum estimators to find from the estimated set a consistent estimator of an appropriately chosen solution. The estimator is asymptotically normal, locally efficient and attains the semi-parametric efficiency bound under certain regularity conditions. We illustrate the proposed approach via simulations and a real data application on home pricing.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkad047","type":"journal-article","created":{"date-parts":[[2023,5,7]],"date-time":"2023-05-07T23:51:36Z","timestamp":1683503496000},"page":"913-935","source":"Crossref","is-referenced-by-count":17,"title":["Non-parametric inference about mean functionals of non-ignorable non-response data without identifying the joint distribution"],"prefix":"10.1093","volume":"85","author":[{"given":"Wei","family":"Li","sequence":"first","affiliation":[{"name":"Center for Applied Statistics and School of Statistics, Renmin University of China , Beijing , P.R. 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