{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,30]],"date-time":"2026-06-30T18:51:47Z","timestamp":1782845507100,"version":"3.54.5"},"reference-count":151,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,18]],"date-time":"2021-07-18T00:00:00Z","timestamp":1626566400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["LabEx PERSYVAL-Lab (ANR-11- 466 LABX-0025-01)"],"award-info":[{"award-number":["LabEx PERSYVAL-Lab (ANR-11- 466 LABX-0025-01)"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we focus on extended informational measures based on a convex function \u03d5: entropies, extended Fisher information, and generalized moments. Both the generalization of the Fisher information and the moments rely on the definition of an escort distribution linked to the (entropic) functional \u03d5. We revisit the usual maximum entropy principle\u2014more precisely its inverse problem, starting from the distribution and constraints, which leads to the introduction of state-dependent \u03d5-entropies. Then, we examine interrelations between the extended informational measures and generalize relationships such the Cram\u00e9r\u2013Rao inequality and the de Bruijn identity in this broader context. In this particular framework, the maximum entropy distributions play a central role. Of course, all the results derived in the paper include the usual ones as special cases.<\/jats:p>","DOI":"10.3390\/e23070911","type":"journal-article","created":{"date-parts":[[2021,7,18]],"date-time":"2021-07-18T21:16:48Z","timestamp":1626643008000},"page":"911","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["\u03d5-Informational Measures: Some Results and Interrelations"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7595-9601","authenticated-orcid":false,"given":"Steeve","family":"Zozor","sequence":"first","affiliation":[{"name":"GIPSA-Lab, CNRS, Grenoble INP,  University Grenoble Alpes, 38000 Grenoble, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Fran\u00e7ois","family":"Bercher","sequence":"additional","affiliation":[{"name":"CNRS, LIGM, University Gustave Eiffel, 77454 Marne-la-Vall\u00e9e, France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,18]]},"reference":[{"key":"ref_1","first-page":"273","article-title":"Thermodynamik quantenmechanischer Gesamtheiten","volume":"1","year":"1927","journal-title":"Nachr. 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