{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,5]],"date-time":"2026-10-05T06:33:32Z","timestamp":1791182012963,"version":"4.1.0"},"reference-count":22,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2015,6,11]],"date-time":"2015-06-11T00:00:00Z","timestamp":1433980800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The paper proposes a new non-parametric density estimator from region-censored observations with application in the context of population studies, where standard maximum likelihood is affected by over-fitting and non-uniqueness problems. It is a maximum entropy estimator that satisfies a set of constraints imposing a close fit to the empirical distributions associated with the set of censoring regions. The degree of relaxation of the data-fit constraints is chosen, such that the likelihood of the inferred model is maximal. In this manner, the estimator is able to overcome the singularity of the non-parametric maximum likelihood estimator and, at the same time, maintains a good fit to the observations. The behavior of the estimator is studied in a simulation, demonstrating its superior performance with respect to the non-parametric maximum likelihood and the importance of carefully choosing the degree of relaxation of the data-fit constraints. In particular, the predictive performance of the resulting estimator is better, which is important when the population analysis is done in the context of risk assessment. We also apply the estimator to real data in the context of the prevention of hyperbaric decompression sickness, where the available observations are formally equivalent to region-censored versions of the variables of interest, confirming that it is a superior alternative to non-parametric maximum likelihood in realistic situations.<\/jats:p>","DOI":"10.3390\/e17063963","type":"journal-article","created":{"date-parts":[[2015,6,11]],"date-time":"2015-06-11T10:44:59Z","timestamp":1434019499000},"page":"3963-3988","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Most Likely Maximum Entropy for Population Analysis with Region-Censored Data"],"prefix":"10.3390","volume":"17","author":[{"given":"Youssef","family":"Bennani","sequence":"first","affiliation":[{"name":"CNRS, Laboratoire I3S-UMR 7271, Universit\u00e9 de Nice-Sophia Antipolis\/CNRS, 06900 Sophia Antipolis, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luc","family":"Pronzato","sequence":"additional","affiliation":[{"name":"CNRS, Laboratoire I3S-UMR 7271, Universit\u00e9 de Nice-Sophia Antipolis\/CNRS, 06900 Sophia Antipolis, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maria","family":"Rendas","sequence":"additional","affiliation":[{"name":"CNRS, Laboratoire I3S-UMR 7271, Universit\u00e9 de Nice-Sophia Antipolis\/CNRS, 06900 Sophia Antipolis, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2015,6,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1080\/01621459.1958.10501452","article-title":"Nonparametric estimation from incomplete observations","volume":"53","author":"Kaplan","year":"1958","journal-title":"J. Am. Stat. Assos."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1111\/j.2517-6161.1976.tb01597.x","article-title":"The empirical distribution function with arbitrarily grouped, censored and truncated data","volume":"38","author":"Turnbull","year":"1976","journal-title":"J. R. Stat. Soc. Ser. B (Methodol.)"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1198\/106186001317114901","article-title":"Computational algorithms for censored-data problems using intersection graphs","volume":"10","author":"Gentleman","year":"2001","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_4","unstructured":"Bennani, Y. (2013). Intersection Graph for Region-Censored Data, I3S."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information theory and statistical mechanics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. 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