{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T11:43:16Z","timestamp":1782301396354,"version":"3.54.5"},"reference-count":47,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,8,26]],"date-time":"2024-08-26T00:00:00Z","timestamp":1724630400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Science and Technology Research Project of Jiangxi Provincial Department of Education","award":["GJJ2200814"],"award-info":[{"award-number":["GJJ2200814"]}]},{"name":"Science and Technology Research Project of Jiangxi Provincial Department of Education","award":["GJJ2203007"],"award-info":[{"award-number":["GJJ2203007"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study investigates the statistical inference of the parameters, reliability function, and hazard function of the generalized Rayleigh distribution under progressive first-failure censoring samples, considering factors such as long product lifetime and challenging experimental conditions. Firstly, the progressive first-failure model is introduced, and the maximum likelihood estimation for the parameters, reliability function, and hazard function under this model are discussed. For interval estimation, confidence intervals have been constructed for the parameters, reliability function, and hazard function using the bootstrap method. Next, in Bayesian estimation, considering informative priors and non-information priors, the Bayesian estimation of the parameters, reliability function, and hazard function under symmetric and asymmetric loss functions is obtained using the MCMC method. Finally, Monte Carlo simulation is conducted to compare mean square errors, evaluating the superiority of the maximum likelihood estimation and Bayesian estimation under different loss functions. The performance of the estimation methods used in the study is illustrated through illustrative examples. The results indicate that Bayesian estimation outperforms maximum likelihood estimation.<\/jats:p>","DOI":"10.3390\/axioms13090580","type":"journal-article","created":{"date-parts":[[2024,8,26]],"date-time":"2024-08-26T10:01:51Z","timestamp":1724666511000},"page":"580","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Estimation of the Reliability Function of the Generalized Rayleigh Distribution under Progressive First-Failure Censoring Model"],"prefix":"10.3390","volume":"13","author":[{"given":"Qin","family":"Gong","sequence":"first","affiliation":[{"name":"College of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rui","family":"Chen","sequence":"additional","affiliation":[{"name":"Business School, Jiangxi University of Science and Technology, Nanchang 330013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haiping","family":"Ren","sequence":"additional","affiliation":[{"name":"Teaching Department of Basic Subjects, Jiangxi University of Science and Technology, Nanchang 330013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fan","family":"Zhang","sequence":"additional","affiliation":[{"name":"Teaching Department of Basic Subjects, Jiangxi University of Science and Technology, Nanchang 330013, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Seong, K., and Lee, K. (2022). Exact likelihood inference for parameter of Exponential distribution under combined Generalized progressive hybrid censoring scheme. Symmetry, 14.","DOI":"10.3390\/sym14091764"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1720033","DOI":"10.1155\/2022\/1720033","article-title":"Estimation using suggested EM algorithm based on progressively Type-II censored samples from a finite mixture of truncated Type-I generalized logistic distributions with an application","volume":"2022","author":"Ateya","year":"2022","journal-title":"Math. Probl. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Alotaibi, R., Nassar, M., and Elshahhat, A. (2023). Estimations of modified Lindley parameters using progressive Type-II censoring with applications. Axioms, 12.","DOI":"10.3390\/axioms12020171"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1080\/00949659508811684","article-title":"Failure-censored reliability sampling plans for the exponential distribution","volume":"52","author":"Balasooriya","year":"1995","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"654","DOI":"10.1080\/00401706.1965.10490310","article-title":"Theory and technique of variation research","volume":"7","author":"Dykstra","year":"1964","journal-title":"Technometrics"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3659","DOI":"10.1016\/j.csda.2009.03.010","article-title":"On estimation based on progressive first-failure censored sampling","volume":"53","author":"Wu","year":"2009","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1352","DOI":"10.1002\/qre.3298","article-title":"Estimation of stress-strength reliability for beta log Weibull distribution using progressive first-failure censored samples","volume":"39","author":"Shi","year":"2023","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Abu-Moussa, M.H., Alsadat, N., and Sharawy, A. (2023). On estimation of reliability functions for the extended Rayleigh distribution under progressive first-failure censoring model. Axioms, 12.","DOI":"10.3390\/axioms12070680"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"22419","DOI":"10.3934\/math.20231144","article-title":"Statistical analysis of progressively first-failure censored data via beta-binomial removals","volume":"8","author":"Elshahhat","year":"2023","journal-title":"Aims Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"29","DOI":"10.3934\/math.2023002","article-title":"Reliability analysis of constant partially accelerated life tests under progressive first-failure Type-II censored data from Lomax model: EM and MCMC algorithms","volume":"8","author":"Eliwa","year":"2023","journal-title":"Aims Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1023\/A:1011352923990","article-title":"Inference for reliability and stress-strength for a scaled Burr type X distribution","volume":"7","author":"Surles","year":"2001","journal-title":"Lifetime Data Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"11523","DOI":"10.1016\/j.aej.2022.05.010","article-title":"A new generalized Rayleigh distribution with analysis to big data of an online community","volume":"61","author":"Shen","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_13","first-page":"6307435","article-title":"Estimations in a constant-stress partially accelerated life test for generalized Rayleigh distribution under Type-II hybrid censoring scheme","volume":"2022","author":"Rabie","year":"2022","journal-title":"J. Mathem."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/s00180-020-01021-y","article-title":"Inference and optimal censoring scheme for progressively Type-II censored competing risks model for generalized Rayleigh distribution","volume":"36","author":"Ren","year":"2021","journal-title":"Comput. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"126704","DOI":"10.1016\/j.neucom.2023.126704","article-title":"ELM parameter estimation in view of maximum likelihood","volume":"557","author":"Yang","year":"2023","journal-title":"Neurocomputing"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2773","DOI":"10.1007\/s00500-022-06766-w","article-title":"Estimating unknown parameters in uncertain differential equation by maximum likelihood estimation","volume":"26","author":"Liu","year":"2022","journal-title":"Soft. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2314","DOI":"10.1155\/2022\/4092576","article-title":"Approximate maximum likelihood estimations for the parameters of the generalized gudermannian distribution and its characterizations","volume":"2022","author":"Rasekhi","year":"2022","journal-title":"J. Math."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Cai, Y.X., and Gui, W.H. (2021). Classical and Bayesian inference for a progressive first-failure censored left-truncated normal distribution. Symmetry, 13.","DOI":"10.3390\/sym13030490"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1016\/S0167-9473(01)00091-3","article-title":"Estimation of parameters from progressively censored data using EM algorithm","volume":"39","author":"Ng","year":"2002","journal-title":"Comput. Stat. Data"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"107552","DOI":"10.1016\/j.csda.2022.107552","article-title":"Bootstrap confidence intervals for multiple change points based on moving sum procedures","volume":"175","author":"Cho","year":"2022","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1093\/jssam\/smaa038","article-title":"Parametric bootstrap confidence intervals for the multivariate Fay\u2013Herriot model","volume":"10","author":"Saegusa","year":"2022","journal-title":"J. Surv. Stat. Methodol."},{"key":"ref_22","first-page":"445","article-title":"Comparison of jackknife and bootstrap methods in estimating confidence intervals","volume":"153","author":"Sroka","year":"2021","journal-title":"Sci. Pap. Silesian Univ. Technol. Organ. Manag. Ser."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Song, X.F., Xiong, Z.Y., and Gui, W.H. (2022). Parameter estimation of exponentiated half-logistic distribution for left-truncated and right-censored data. Mathematics, 10.","DOI":"10.3390\/math10203838"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7718","DOI":"10.1080\/03610926.2022.2056752","article-title":"Take a look at the hierarchical Bayesian estimation of parameters from several different angles","volume":"52","author":"Han","year":"2022","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"223","DOI":"10.3233\/IDA-216283","article-title":"Bayesian estimation of decay parameters in Hawkes processes","volume":"27","author":"Tiago","year":"2023","journal-title":"Intell. Data Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4455","DOI":"10.3934\/mbe.2023207","article-title":"Bayesian parameter estimation for phosphate dynamics during hemodialysis","volume":"20","author":"Bangsgaard","year":"2023","journal-title":"Math. Biosci. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"023021","DOI":"10.1103\/PhysRevD.108.023021","article-title":"Bayesian parameter estimation on boson-star binary signals with a coherent inspiral template and spin-dependent quadrupolar corrections","volume":"108","author":"Vaglio","year":"2023","journal-title":"Phys. Rev. D"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"324","DOI":"10.1080\/00949655.2015.1007986","article-title":"A study of the effect of loss functions on the Bayes estimates of dynamic cumulative residual entropy for Pareto distribution under upper record values","volume":"86","author":"Renjini","year":"2016","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"284","DOI":"10.3844\/jmssp.2017.284.291","article-title":"On Bayesian premium estimators for gamma lindley model under squared error loss function and linex loss function","volume":"13","author":"Sadoun","year":"2017","journal-title":"J. Math. Stat."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3952","DOI":"10.1080\/03610926.2014.912057","article-title":"Bayesian estimation for non zero inflated modified power series distribution under linex and generalized entropy loss functions","volume":"45","author":"Malgorzata","year":"2016","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_31","unstructured":"Wasan, M.T. (1970). Parametric Estimation, McGraw-Hill Book Company."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1109\/24.536992","article-title":"The use of precautionary loss functions in risk analysis","volume":"45","author":"Norstrom","year":"1996","journal-title":"IEEE Trans. Reliab. Theory"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"705","DOI":"10.1016\/j.apm.2023.01.009","article-title":"A note on the posterior risk of the entropy loss function","volume":"117","author":"Han","year":"2023","journal-title":"Appl. Math. Model."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Abdel-Aty, Y., Kayid, M., and Alomani, G. (2023). Generalized Bayes estimation based on a joint Type-II censored sample from K-exponential populations. Mathematics, 11.","DOI":"10.20944\/preprints202306.1294.v1"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Ren, H.P., Gong, Q., and Hu, X. (2023). Estimation of entropy for generalized Rayleigh distribution under progressively Type-II censored samples. Axioms, 12.","DOI":"10.3390\/axioms12080776"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1971","DOI":"10.1080\/03610918.2019.1589510","article-title":"E-Bayesian estimations of parameter and its evaluation standard: E-MSE (expected mean square error) under different loss functions","volume":"50","author":"Han","year":"2021","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"6068","DOI":"10.1016\/j.apm.2012.12.008","article-title":"A study of the effect of the loss function on Bayes estimate, posterior risk and hazard function for lindley distribution","volume":"37","author":"Ali","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_38","first-page":"32","article-title":"Parameter estimation of buffer autoregressive models based on Bayesian inference","volume":"12","author":"Wei","year":"2023","journal-title":"Stat. Appl."},{"key":"ref_39","first-page":"57","article-title":"Parameter estimation of logistic regression model based on MCMC algorithm\u2014A case study of smart sleep bracelet","volume":"28","author":"Zhao","year":"2022","journal-title":"Mod. Comput."},{"key":"ref_40","first-page":"563","article-title":"Estimation in Weibull distribution under progressively Type-I hybrid censored data","volume":"20","author":"Asar","year":"2023","journal-title":"REVSTAT-Stat. J."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1111\/stan.12281","article-title":"Bayesian inference for a mixture double autoregressive model","volume":"77","author":"Yang","year":"2023","journal-title":"Stat. Neerlandica"},{"key":"ref_42","first-page":"1580","article-title":"Semi-parametric hierarchical Bayesian modeling and optimization","volume":"45","author":"Chen","year":"2023","journal-title":"Syst. Eng. Electron."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Wang, X.Y., and Gui, W.H. (2021). Bayesian estimation of entropy for burr type XII distribution under progressive Type-II censored data. Mathematics, 9.","DOI":"10.3390\/math9040313"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1080\/00031305.1995.10476150","article-title":"A simple simulational algorithm for generating progressive Type-II censored samples","volume":"49","author":"Balakrishnan","year":"1995","journal-title":"Am. Stat."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1093\/oxfordjournals.aje.a120129","article-title":"Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli","volume":"72","author":"Bjerkedal","year":"1960","journal-title":"Am. J. Epidemiol."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1080\/02664760802698961","article-title":"On the comparison of the Fisher information of the log-normal and generalized Rayleigh distributions","volume":"37","author":"Alshunnar","year":"2010","journal-title":"J. Appl. Stat."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"3994361","DOI":"10.1155\/2022\/3994361","article-title":"A New version of Weighte Weibull distribution: Modelling to COVID-19 data","volume":"2022","author":"Alahmadi","year":"2022","journal-title":"Discret. Dyn. Nat. Soc."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/580\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:43:01Z","timestamp":1760110981000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/9\/580"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,26]]},"references-count":47,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["axioms13090580"],"URL":"https:\/\/doi.org\/10.3390\/axioms13090580","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,26]]}}}