{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,30]],"date-time":"2026-06-30T16:17:34Z","timestamp":1782836254910,"version":"3.54.5"},"reference-count":21,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,8]],"date-time":"2020-06-08T00:00:00Z","timestamp":1591574400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004054","name":"King Abdulaziz University","doi-asserted-by":"publisher","award":["RG-12-130-41"],"award-info":[{"award-number":["RG-12-130-41"]}],"id":[{"id":"10.13039\/501100004054","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, E-Bayesian estimation of the scale parameter, reliability and hazard rate functions of Chen distribution are considered when a sample is obtained from a type-I censoring scheme. The E-Bayesian estimators are obtained based on the balanced squared error loss function and using the gamma distribution as a conjugate prior for the unknown scale parameter. Also, the E-Bayesian estimators are derived using three different distributions for the hyper-parameters. Some properties of E-Bayesian estimators based on balanced squared error loss function are discussed. A simulation study is performed to compare the efficiencies of different estimators in terms of minimum mean squared errors. Finally, a real data set is analyzed to illustrate the applicability of the proposed estimators.<\/jats:p>","DOI":"10.3390\/e22060636","type":"journal-article","created":{"date-parts":[[2020,6,9]],"date-time":"2020-06-09T04:19:39Z","timestamp":1591676379000},"page":"636","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["E-Bayesian Estimation of Chen Distribution Based on Type-I Censoring Scheme"],"prefix":"10.3390","volume":"22","author":[{"given":"Ali","family":"Algarni","sequence":"first","affiliation":[{"name":"Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudia Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Abdullah M.","family":"Almarashi","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudia Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5781-5364","authenticated-orcid":false,"given":"Hassan","family":"Okasha","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudia Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4685-2199","authenticated-orcid":false,"given":"Hon Keung Tony","family":"Ng","sequence":"additional","affiliation":[{"name":"Department of Statistical Science, Southern Methodist University, Dallas, TX 75275-0332, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/S0951-8320(02)00022-4","article-title":"A modified Weibull extension with bathtub-shaped failure rate function","volume":"76","author":"Xie","year":"2002","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1109\/TR.2002.805788","article-title":"A modified Weibull distribution","volume":"52","author":"Lai","year":"2003","journal-title":"IEEE Trans. Reliab."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"10236","DOI":"10.1080\/03610926.2016.1231816","article-title":"Alpha Power Weibull Distribution: Properties and Applications","volume":"46","author":"Nassar","year":"2017","journal-title":"Commun. Stat.-Theory And Methods"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/j.cam.2017.12.001","article-title":"A new extension of Weibull distribution: Properties and different methods of estimation","volume":"336","author":"Nassar","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2389","DOI":"10.1080\/02664763.2012.710899","article-title":"Estimating the parameters of a bathtub-shaped distribution under progressive type-II censoring","volume":"39","author":"Rastogi","year":"2012","journal-title":"J. Appl. Stat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/S0167-7152(00)00044-4","article-title":"A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function","volume":"49","author":"Chen","year":"2000","journal-title":"Stat. Probabil Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.1080\/02664760802264996","article-title":"Estimation of the two parameter bathtub-shaped lifetime distribution with progressive censoring","volume":"35","author":"Wu","year":"2008","journal-title":"J. Appl. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1080\/02664763.2013.847907","article-title":"Bayesian estimation based on progressive Type-II censoring from two-parameter bathtub-shaped lifetime model: an Markov chain Monte Carlo approach","volume":"41","author":"Ahmed","year":"2014","journal-title":"J. Appl. Stat."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1080\/00949655.2016.1209199","article-title":"Estimation and prediction for Chen distribution with bathtub shape under progressive censoring","volume":"87","author":"Kayal","year":"2017","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1180","DOI":"10.1016\/j.jspi.2008.07.008","article-title":"Bayes estimation based on k-record data from a general class of distributions under balanced type loss functions","volume":"139","author":"Ahmadi","year":"2009","journal-title":"J. Stat. Plan. Infer."},{"key":"ref_11","first-page":"31","article-title":"The structure of hierarchical prior distribution and its applications","volume":"6","author":"Han","year":"1997","journal-title":"Chin. Oper. Res. Manag. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1915","DOI":"10.1016\/j.apm.2008.03.019","article-title":"E-Bayesian estimation and hierarchical Bayesian estimation of failure rate","volume":"33","author":"Han","year":"2009","journal-title":"Appl. Math. Model."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4730","DOI":"10.1016\/j.apm.2011.03.055","article-title":"E-Bayesian estimation for the Burr type XII model based on type-2 censoring","volume":"35","author":"Jaheen","year":"2011","journal-title":"Appl. Math. Model."},{"key":"ref_14","first-page":"33","article-title":"E-Bayesian Estimation of System Reliability with Weibull Distribution Based on Type-2 Censoring","volume":"4","author":"Okasha","year":"2012","journal-title":"J. Adv. Res. Sci. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/j.joems.2013.12.009","article-title":"E-Bayesian estimation for the Lomax distribution based on type-II censored data","volume":"22","author":"Okasha","year":"2014","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"236","DOI":"10.2991\/jsta.d.190820.001","article-title":"Estimation for the Exponential Model Based on Record Statistic","volume":"18","author":"Okasha","year":"2019","journal-title":"J. Stat. Theory Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"658","DOI":"10.1016\/j.apm.2015.05.004","article-title":"E-Bayesian estimation for the geometric model based on record statistics","volume":"40","author":"Okasha","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1080\/00949655.2016.1221408","article-title":"The E-Bayesian and hierarchical Bayesian estimations of Pareto distribution parameter under different loss functions","volume":"87","author":"Ming","year":"2017","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_19","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (1980). Tables of Integrals, Series and Products (Corrected and Enlarged Edition), Academic Press."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Hand, D.J., Daly, F., Lunn, A.D., and McConway, K.J. (1994). A Handbook of Small Data Sets, Chapman & Hall.","DOI":"10.1007\/978-1-4899-7266-8"},{"key":"ref_21","unstructured":"Kayal, T., Tripathi, Y.M., Kundu, D., and Rastogi, M. (2019). Statistical inference of Chen distribution based on Type-I progressive hybrid censored samples. Stat. Optim. Inf. Comput., Available online: https:\/\/pdfs.semanticscholar.org\/4d38\/8e0ea99ca778493acefd31eb146ee50ca619.pdf."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/6\/636\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:36:52Z","timestamp":1760175412000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/6\/636"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,6,8]]},"references-count":21,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2020,6]]}},"alternative-id":["e22060636"],"URL":"https:\/\/doi.org\/10.3390\/e22060636","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,6,8]]}}}