{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T07:25:56Z","timestamp":1767857156022,"version":"3.49.0"},"reference-count":40,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,9,26]],"date-time":"2021-09-26T00:00:00Z","timestamp":1632614400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, a discrete analogue of an extension to a two-parameter half-logistic model is proposed for modeling count data. The probability mass function of the new model can be expressed as a mixture representation of a geometric model. Some of its statistical properties, including hazard rate function, moments, moment generating function, conditional moments, stress-strength analysis, residual entropy, cumulative residual entropy and order statistics with its moments, are derived. It is found that the new distribution can be utilized to model positive skewed data, and it can be used for analyzing equi- and over-dispersed data. Furthermore, the hazard rate function can be either decreasing, increasing or bathtub. The parameter estimation through the classical point of view has been performed using the method of maximum likelihood. A detailed simulation study is carried out to examine the outcomes of the estimators. Finally, two distinctive real data sets are analyzed to prove the flexibility of the proposed discrete distribution.<\/jats:p>","DOI":"10.3390\/sym13101790","type":"journal-article","created":{"date-parts":[[2021,9,27]],"date-time":"2021-09-27T23:08:31Z","timestamp":1632784111000},"page":"1790","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Probability Mass Function for Various Shapes of the Failure Rates, Asymmetric and Dispersed Data with Applications to Coronavirus and Kidney Dysmorphogenesis"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7585-5519","authenticated-orcid":false,"given":"Mahmoud","family":"El-Morshedy","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6638-2185","authenticated-orcid":false,"given":"Morad","family":"Alizadeh","sequence":"additional","affiliation":[{"name":"Department of Statistics, Persian Gulf University, Bushehr 75169, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Afrah","family":"Al-Bossly","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5619-210X","authenticated-orcid":false,"given":"Mohamed","family":"Eliwa","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1080\/00949658508810784","article-title":"Order statistics from the half logistic distribution","volume":"20","author":"Balakrishnan","year":"1985","journal-title":"J. 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