{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T08:12:08Z","timestamp":1778746328956,"version":"3.51.4"},"reference-count":41,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,1]],"date-time":"2022-10-01T00:00:00Z","timestamp":1664582400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11861049"],"award-info":[{"award-number":["11861049"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022MS01006"],"award-info":[{"award-number":["2022MS01006"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004763","name":"Natural Science Foundation of Inner Mongolia","doi-asserted-by":"publisher","award":["11861049"],"award-info":[{"award-number":["11861049"]}],"id":[{"id":"10.13039\/501100004763","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004763","name":"Natural Science Foundation of Inner Mongolia","doi-asserted-by":"publisher","award":["2022MS01006"],"award-info":[{"award-number":["2022MS01006"]}],"id":[{"id":"10.13039\/501100004763","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we discuss the statistical analysis of a simple step-stress accelerated competing failure model under progressively Type-II censoring. It is assumed that there is more than one cause of failure, and the lifetime of the experimental units at each stress level follows exponential distribution. The distribution functions under different stress levels are connected through the cumulative exposure model. The maximum likelihood, Bayesian, Expected Bayesian, and Hierarchical Bayesian estimations of the model parameters are derived based on the different loss function. Based on Monte Carlo Simulations. We also get the average length and the coverage probability of the 95% confidence intervals and highest posterior density credible intervals of the parameters. From the numerical studies, it can be seen that the proposed Expected Bayesian estimations and Hierarchical Bayesian estimations have better performance in terms of the average estimates and mean squared errors, respectively. Finally, the methods of statistical inference discussed here are illustrated with a numerical example.<\/jats:p>","DOI":"10.3390\/e24101405","type":"journal-article","created":{"date-parts":[[2022,10,8]],"date-time":"2022-10-08T04:04:56Z","timestamp":1665201896000},"page":"1405","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["E-Bayesian and H-Bayesian Inferences for a Simple Step-Stress Model with Competing Failure Model under Progressively Type-II Censoring"],"prefix":"10.3390","volume":"24","author":[{"given":"Ying","family":"Wang","sequence":"first","affiliation":[{"name":"College of Science, Inner Mongolia University of Technology, Hohhot 010051, China"},{"name":"School of Statistics and Mathematics, Inner Mongolia University of Finance and Economics, Hohhot 010070, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5889-1326","authenticated-orcid":false,"given":"Zaizai","family":"Yan","sequence":"additional","affiliation":[{"name":"College of Science, Inner Mongolia University of Technology, Hohhot 010051, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yan","family":"Chen","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Statistics, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1080\/00401706.1978.10489643","article-title":"Theory for optimum accelerated censored life tests for Weibull and extreme value distribution","volume":"20","author":"Nelson","year":"1978","journal-title":"Technometrics"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Nelson, W.B. 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