{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T07:49:07Z","timestamp":1762069747414,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T00:00:00Z","timestamp":1665446400000},"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>This paper introduces and studies a new generalization of cumulative past extropy called weighted cumulative past extropy (WCPJ) for continuous random variables. We explore the following: if the WCPJs of the last order statistic are equal for two distributions, then these two distributions will be equal. We examine some properties of the WCPJ, and a number of inequalities involving bounds for WCPJ are obtained. Studies related to reliability theory are discussed. Finally, the empirical version of the WCPJ is considered, and a test statistic is proposed. The critical cutoff points of the test statistic are computed numerically. Then, the power of this test is compared to a number of alternative approaches. In some situations, its power is superior to the rest, and in some other settings, it is somewhat weaker than the others. The simulation study shows that the use of this test statistic can be satisfactory with due attention to its simple form and the rich information content behind it.<\/jats:p>","DOI":"10.3390\/e24101444","type":"journal-article","created":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T06:13:27Z","timestamp":1665468807000},"page":"1444","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Weighted Cumulative Past Extropy and Its Inference"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9957-6963","authenticated-orcid":false,"given":"Mohammad Reza","family":"Kazemi","sequence":"first","affiliation":[{"name":"Department of Statistics, Faculty of Science, Fasa University, Fasa IS74, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Majid","family":"Hashempour","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Hormozgan, Bandar Abbas 79177, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6772-5167","authenticated-orcid":false,"given":"Maria","family":"Longobardi","sequence":"additional","affiliation":[{"name":"Dipartimento di Biologia, Universit\u00e0 degli Studi di Napoli Federico II, 80126 Naples, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. 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