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When the input distribution is known, we investigate the already existing estimator defined in [E. Song, B. L. Nelson, and J. Staum, SIAM\/ASA J. Uncertain. Quantif., 4 (2016), pp. 1060--1083] and suggest a new one with a lower variance. Then, when the distribution of the inputs is unknown, we extend these estimators. We provide asymptotic properties of the estimators studied in this article. We also apply one of these estimators to a real data set.<\/jats:p>","DOI":"10.1137\/18m1234631","type":"journal-article","created":{"date-parts":[[2020,5,11]],"date-time":"2020-05-11T14:47:09Z","timestamp":1589208429000},"page":"693-716","source":"Crossref","is-referenced-by-count":43,"title":["Variance Reduction for Estimation of Shapley Effects and Adaptation to Unknown Input Distribution"],"prefix":"10.1137","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1333-2149","authenticated-orcid":true,"given":"Baptiste","family":"Broto","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5336-5714","authenticated-orcid":true,"given":"Fran\u00e7ois","family":"Bachoc","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marine","family":"Depecker","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,5,11]]},"reference":[{"key":"atypb1","unstructured":"F. Bachoc, F. Gamboa, J.M. Loubes, and L. Risser,\n                      Entropic Variable Boosting for Explainability and Interpretability in Machine Learning\n                      , preprint,http:\/\/arxiv.org\/abs\/1810.07924, 2018."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1051\/proc\/201965266"},{"key":"atypb3","unstructured":"T. B. Berrett and R. J. Samworth,\n                      Efficient Two-Sample Functional Estimation and the Super-Oracle Phenomenon\n                      , preprint,http:\/\/arxiv.org\/abs\/1904.09347, 2019 (accessed 2019-09-09)."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1214\/18-AOS1688"},{"key":"atypb5","first-page":"285","volume":"49","author":"Besse P.","year":"2007","journal-title":"Pollution Atmosph\u00e9rique"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1111\/risa.12555"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.matcom.2019.02.008"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/j.cor.2008.04.004"},{"key":"atypb9","unstructured":"G. Chastaing,\n                      Indices de Sobol g\u00e9n\u00e9ralis\u00e9s pour variables d\u00e9pendantes\n                      , Ph.D. thesis, Universit\u00e9 de Grenoble, 2013,https:\/\/tel.archives-ouvertes.fr\/tel-00930229\/document(accessed 2017-03-27)."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/s11009-016-9540-5"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1016\/j.artint.2008.05.003"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2014.07.018"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1214\/14-EJS895"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1080\/02331888.2015.1105803"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1214\/aos\/1013203453"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/0951-8320(96)00002-6"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1111\/j.1467-6451.2011.00447.x"},{"key":"atypb18","unstructured":"B. Iooss and C. Prieur,\n                      Shapley Effects for Sensitivity Analysis with Dependent Inputs: Comparisons with Sobol' Indices, Numerical Estimation and Applications\n                      , preprint,http:\/\/arxiv.org\/abs\/1707.01334, 2017 (accessed 2017-11-28)."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2005.11.047"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1051\/ps\/2013040"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1023\/A:1008306431147"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2011.08.008"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1186\/1471-2105-9-361"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/130936233"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/16M1097717"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2009.11.005"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2012.11.047"},{"key":"atypb28","first-page":"1135","author":"Ribeiro M. T.","year":"2016","journal-title":"ACM"},{"key":"atypb29","doi-asserted-by":"crossref","unstructured":"L. S. Shapley,\n                      A value for $n$-person games\n                      , in Contribution to the Theory of Games, Vol. 2, Ann. Math. Stud. 28, Princeton University Press, 1953, pp. 307-317.","DOI":"10.1515\/9781400881970-018"},{"key":"atypb30","first-page":"407","volume":"1","author":"Sobol I. M.","year":"1993","journal-title":"Math. Model. Comput. Experiments"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1137\/15M1048070"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1080\/10485252.2013.784762"},{"key":"atypb33","unstructured":"B. J. Williams, T. J. Santner, and W. I. Notz,\n                      The Design and Analysis of Computer Experiments\n                      , Vol. 1, Springer, 2003."}],"container-title":["SIAM\/ASA Journal on Uncertainty Quantification"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1234631","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:22:20Z","timestamp":1787340140000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1234631"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["10.1137\/18M1234631"],"URL":"https:\/\/doi.org\/10.1137\/18m1234631","relation":{},"ISSN":["2166-2525"],"issn-type":[{"value":"2166-2525","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1]]}}}