{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T01:36:57Z","timestamp":1768354617213,"version":"3.49.0"},"reference-count":46,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,19]],"date-time":"2023-06-19T00:00:00Z","timestamp":1687132800000},"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 order to represent the data with non-monotonic failure rates and produce a better fit, a novel distribution is created in this study using the alpha power family of distributions. This distribution is called the alpha-power Kum-modified size-biased Lehmann type II or, in short, the AP-Kum-MSBL-II distribution. This distribution is established for modeling bounded data in the interval (0,1). The proposed distribution\u2019s moment-generating function, mode, quantiles, moments, and stress\u2013strength reliability function are obtained, among other attributes. To estimate the parameters of the proposed distribution, estimation methods such as the maximum likelihood method and Bayesian method are employed to estimate the unknown parameters for the AP-Kum-MSBL-II distribution. Moreover, the confidence intervals, credible intervals, and coverage probability are calculated for all parameters. The symmetric and asymmetric loss functions are used to find the Bayesian estimators using the Markov chain Monte Carlo (MCMC) method. Furthermore, the proposed distribution\u2019s usefulness is demonstrated using three real data sets. One of them is a medical data set dealing with COVID-19 patients\u2019 mortality rate, the second is a trade share data set, and the third is from the engineering area, as well as extensive simulated data, which were applied to assess the performance of the estimators of the proposed distribution.<\/jats:p>","DOI":"10.3390\/sym15061283","type":"journal-article","created":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T01:37:38Z","timestamp":1687225058000},"page":"1283","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["The Unit Alpha-Power Kum-Modified Size-Biased Lehmann Type II Distribution: Theory, Simulation, and Applications"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5950-7899","authenticated-orcid":false,"given":"Rabab S.","family":"Gomaa","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8035-7653","authenticated-orcid":false,"given":"Alia M.","family":"Magar","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8884-8281","authenticated-orcid":false,"given":"Najwan","family":"Alsadat","sequence":"additional","affiliation":[{"name":"Department of Quantitative Analysis, College of Business Administration, King Saud University, P.O. Box 71115, Riyadh 11587, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3888-1275","authenticated-orcid":false,"given":"Ehab M.","family":"Almetwally","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Business Administration, Delta University for Science and Technology, Gamasa 11152, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8938-9187","authenticated-orcid":false,"given":"Ahlam H.","family":"Tolba","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Alsadat, N., Hassan, A.S., Elgarhy, M., Chesneau, C., and Mohamed, R.E. (2023). An Efficient Stress\u2013Strength Reliability Estimate of the Unit Gompertz Distribution Using Ranked Set Sampling. 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