{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,14]],"date-time":"2026-02-14T07:15:43Z","timestamp":1771053343620,"version":"3.50.1"},"reference-count":32,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,23]],"date-time":"2025-03-23T00:00:00Z","timestamp":1742688000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU)","award":["IMSIU-DDRSP2503"],"award-info":[{"award-number":["IMSIU-DDRSP2503"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper explores a progressive-stress accelerated life test under progressive type-II censoring with binomial random removal. It assumes a cumulative exposure model in which the lifetimes of test units follow a Marshall\u2013Olkin length-biased exponential distribution. The study derives maximum likelihood and Bayes estimates of the model parameters and constructs Bayes estimates of the unknown parameters under various loss functions. In addition, this study provides approximate, credible, and bootstrapping confidence intervals for the estimators. Moreover, it evaluates three optimal test methods to determine the most effective censoring approach based on various optimality criteria. A real-life dataset is analyzed to demonstrate the proposed procedures and simulation studies used to compare two different designs of the progressive-stress test.<\/jats:p>","DOI":"10.3390\/axioms14040244","type":"journal-article","created":{"date-parts":[[2025,3,24]],"date-time":"2025-03-24T06:21:38Z","timestamp":1742797298000},"page":"244","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Inference Based on Progressive-Stress Accelerated Life-Testing for Extended Distribution via the Marshall-Olkin Family Under Progressive Type-II Censoring with Optimality Techniques"],"prefix":"10.3390","volume":"14","author":[{"given":"Ehab M.","family":"Almetwally","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia"}]},{"given":"Osama M.","family":"Khaled","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Science, Port Said University, Port Said 42521, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6212-432X","authenticated-orcid":false,"given":"Haroon M.","family":"Barakat","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44759, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,23]]},"reference":[{"key":"ref_1","unstructured":"Nelson, W.B. (2009). Accelerated Testing: Statistical Models, Test Plans, and Data Analysis, John Wiley & Sons."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1002\/qre.2562","article-title":"Statistical inference on progressive-stress accelerated life testing for the logistic exponential distribution under progressive type-II censoring","volume":"36","author":"Mahto","year":"2020","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1109\/TR.1987.5222320","article-title":"Some aspects of accelerated life-testing by progressive stress","volume":"36","author":"Yin","year":"1987","journal-title":"IEEE Trans. Reliab."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1109\/24.159808","article-title":"Optimum simple ramp-tests for the Weibull distribution and type-I censoring","volume":"41","author":"Bai","year":"1992","journal-title":"IEEE Trans. Reliab."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1007\/s00184-014-0526-4","article-title":"One-sample Bayesian prediction intervals based on progressively type-II censored data from the half-logistic distribution under progressive stress model","volume":"78","author":"Hashem","year":"2015","journal-title":"Metrika"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Balakrishnan, N., and Aggarwala, R. (2000). Progressive Censoring: Theory, Methods, and Applications, Springer Science & Business Media.","DOI":"10.1007\/978-1-4612-1334-5"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Balakrishnan, N., and Cramer, E. (2014). The Art of Progressive Censoring: Applications to Reliability and Quality, Springer. Statistics for Industry and Technology.","DOI":"10.1007\/978-0-8176-4807-7"},{"key":"ref_8","first-page":"237","article-title":"Statistical inference of Weibull distribution for tampered failure rate model in progressive stress accelerated life-testing","volume":"14","author":"Heliang","year":"2004","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1007\/s00184-006-0106-3","article-title":"Progressive stress accelerated life tests under finite mixture models","volume":"66","year":"2007","journal-title":"Metrika"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5259","DOI":"10.1016\/j.cam.2011.05.035","article-title":"Inference for a progressive stress model from Weibull distribution under progressive type-II censoring","volume":"235","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2483","DOI":"10.1002\/qre.2212","article-title":"lassical and Bayesian inference on progressive-stress accelerated life-testing for the extension of the exponential distribution under progressive type-II censoring","volume":"33","author":"Ali","year":"2017","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1016\/j.aej.2024.05.045","article-title":"Bayesian estimation and prediction under progressive-stress accelerated life test for a log-logistic model","volume":"101","author":"Mahto","year":"2024","journal-title":"Alex. Eng. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"425","DOI":"10.3934\/math.2022028","article-title":"Inference on a new distribution under progressive-stress accelerated life tests and progressive type-II censoring based on a series-parallel system","volume":"7","author":"Abushal","year":"2022","journal-title":"AIMS Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"101994","DOI":"10.1016\/j.jksus.2022.101994","article-title":"Progressive stress accelerated life test for inverse Weibull failure model: A parametric inference","volume":"34","author":"Ismail","year":"2022","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"9558650","DOI":"10.1155\/2022\/9558650","article-title":"Single and multiple ramp progressive stress with binomial removal: Practical application for industry","volume":"2022","author":"Hussam","year":"2022","journal-title":"Math. Probl. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Alotaibi, R., Alamri, F.S., Almetwally, E.M., Wang, M., and Rezk, H. (2022). Classical and bayesian inference of a progressive-stress model for the Nadarajah-Haghighi distribution with type II progressive censoring and different loss functions. Mathematics, 10.","DOI":"10.3390\/math10091602"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/j.jksus.2017.09.006","article-title":"The Marshall-Olkin length-biased exponential distribution and its applications","volume":"31","author":"Usman","year":"2019","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2030","DOI":"10.1080\/03610920802192505","article-title":"Estimating the parameters of the generalized exponential distribution in presence of hybrid censoring","volume":"38","author":"Kundu","year":"2009","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_19","first-page":"509","article-title":"Estimation, Prediction and life-testing Plan for the Exponentiated Gumbel Type-II Progressive Censored Data","volume":"21","author":"Maiti","year":"2023","journal-title":"REVSTAT Stat. J."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"782","DOI":"10.1109\/TR.2015.2494370","article-title":"Bayesian estimation and prediction for a hybrid censored lognormal distribution","volume":"65","author":"Singh","year":"2015","journal-title":"IEEE Trans. Reliab."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3669","DOI":"10.1080\/03610918.2019.1630431","article-title":"Estimation of parameters and reliability characteristics for a generalized Rayleigh distribution under progressive type-II censored sample","volume":"50","author":"Maiti","year":"2021","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_22","first-page":"85","article-title":"Bayesian and Non-Bayesian Estimation for the Shape Parameters of New Versions of Bivariate Inverse Weibull Distribution based on Progressive Type II Censoring","volume":"3","author":"Muhammed","year":"2024","journal-title":"Comput. J. Math. Stat. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1276","DOI":"10.1007\/s13198-019-00875-w","article-title":"Estimation for the generalized Fr\u00e9chet distribution under progressive censoring scheme","volume":"10","author":"Maiti","year":"2019","journal-title":"Int. J. Syst. Assur. Eng. Manag."},{"key":"ref_24","first-page":"177","article-title":"Inference for Generalized Progressive Hybrid Type-II Censored Weibull Lifetimes Under Competing Risk Data","volume":"3","year":"2024","journal-title":"Comput. J. Math. Stat. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1033","DOI":"10.1080\/02664760050173355","article-title":"Statistical analysis of Weibull distributed lifetime data under Type II progressive censoring with binomial removals","volume":"27","author":"Tse","year":"2000","journal-title":"J. Appl. Stat."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1007\/s11749-008-0110-1","article-title":"On progressively censored generalized exponential distribution","volume":"18","author":"Pradhan","year":"2009","journal-title":"Test"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Efron, B. (1992). Bootstrap methods: Another look at the jackknife. Breakthroughs in Statistics, Springer.","DOI":"10.1007\/978-1-4612-4380-9_41"},{"key":"ref_28","first-page":"545","article-title":"Bootstrapping Order Statistics with Variable Rank: Accepted-December 2022","volume":"22","author":"Sobh","year":"2022","journal-title":"REVSTAT Stat. J."},{"key":"ref_29","first-page":"923","article-title":"Bootstrap method for central and intermediate order statistics under power normalization","volume":"51","author":"Barakat","year":"2015","journal-title":"Kybernetika"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1080\/03610918.2013.805051","article-title":"Bootstrap method for order statistics and modeling study of the air pollution","volume":"44","author":"Barakat","year":"2015","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_31","unstructured":"Barakat, H.M., Nigm, E.M., and Khaled, O.M. (2019). Statistical Techniques for Modelling Extreme Value Data and Related Applications, Cambridge Scholars Publishing. [1st ed.]."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1647","DOI":"10.1016\/j.jspi.2007.05.042","article-title":"On optimality of extremal schemes in progressive type II censoring","volume":"138","author":"Burkschat","year":"2008","journal-title":"J. Stat. Plan. Inference"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/244\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:58:46Z","timestamp":1760029126000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/4\/244"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,23]]},"references-count":32,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["axioms14040244"],"URL":"https:\/\/doi.org\/10.3390\/axioms14040244","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,3,23]]}}}