{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:16:22Z","timestamp":1760148982854,"version":"build-2065373602"},"reference-count":31,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,6,26]],"date-time":"2023-06-26T00:00:00Z","timestamp":1687737600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University","award":["RSP2023R464"],"award-info":[{"award-number":["RSP2023R464"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In parallel with the concept of R\u00e9nyi entropy for residual lifetime distributions, the R\u00e9nyi entropy of inactivity time of lifetime distributions belonging to asymmetric distributions is a useful measure of independent interest. For a system that turns out to be inactive in time t, the past entropy is considered as an uncertainty measure for the past lifetime distribution. In this study, we consider a coherent system that includes n components and has the property that all the components of the system have failed at time t. To assess the predictability of the coherent system\u2019s lifetime, we use the system\u2019s signature to determine the R\u00e9nyi entropy of its past lifetime. We study several analytical results, including expressions, bounds, and order properties for this measure.<\/jats:p>","DOI":"10.3390\/sym15071310","type":"journal-article","created":{"date-parts":[[2023,6,27]],"date-time":"2023-06-27T01:42:02Z","timestamp":1687830122000},"page":"1310","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["R\u00e9nyi Entropy for Past Lifetime Distributions with Application in Inactive Coherent Systems"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6363-1246","authenticated-orcid":false,"given":"Mohamed","family":"Kayid","sequence":"first","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3456-8393","authenticated-orcid":false,"given":"Mansour","family":"Shrahili","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"202","DOI":"10.2307\/3214930","article-title":"New partial ordering of survival functions based on the notion of uncertainty","volume":"32","author":"Ebrahimi","year":"1995","journal-title":"J. Appl. Probab."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1780","DOI":"10.1109\/TSP.2002.1011217","article-title":"An error-entropy minimization algorithm for supervised training of nonlinear adaptive systems","volume":"50","author":"Erdogmus","year":"2002","journal-title":"IEEE Trans. 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