{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T02:19:52Z","timestamp":1773886792358,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,7]],"date-time":"2022-05-07T00:00:00Z","timestamp":1651881600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Majmaah University","award":["R-2022-110"],"award-info":[{"award-number":["R-2022-110"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper deals with the estimation of the parameters for asymmetric distribution and some lifetime indices such as reliability and hazard rate functions based on progressive first-failure censoring. Maximum likelihood, bootstrap and Bayesian approaches of the distribution parameters and reliability characteristics are investigated. Furthermore, the approximate confidence intervals and highest posterior density credible intervals of the parameters are constructed based on the asymptotic distribution of the maximum likelihood estimators and Markov chain Monte Carlo technique, respectively. In addition, the delta method is implemented to obtain the variances of the reliability and hazard functions. Moreover, we apply two methods of bootstrap to construct the confidence intervals. The Bayes inference based on the squared error and LINEX loss functions is obtained. Extensive simulation studies are conducted to evaluate the behavior of the proposed methods. Finally, a real data set of the COVID-19 mortality rate is analyzed to illustrate the estimation methods developed here.<\/jats:p>","DOI":"10.3390\/sym14050956","type":"journal-article","created":{"date-parts":[[2022,5,8]],"date-time":"2022-05-08T23:27:25Z","timestamp":1652052445000},"page":"956","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Bayesian Inferential Approaches and Bootstrap for the Reliability and Hazard Rate Functions under Progressive First-Failure Censoring for Coronavirus Data from Asymmetric Model"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7828-3536","authenticated-orcid":false,"given":"Rashad M.","family":"EL-Sagheer","sequence":"first","affiliation":[{"name":"Mathematics Department, Faculty of Science, Al-Azhar University, Cairo 11884, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9011-835X","authenticated-orcid":false,"given":"Muqrin A.","family":"Almuqrin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science in Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7585-5519","authenticated-orcid":false,"given":"Mahmoud","family":"El-Morshedy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5619-210X","authenticated-orcid":false,"given":"Mohamed S.","family":"Eliwa","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operation Research, College of Science, Qassim University, P.O. Box 6644, Buraydah 51482, Saudi Arabia"},{"name":"Department of Statistics and Computer Science, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fathy H.","family":"Eissa","sequence":"additional","affiliation":[{"name":"College of Science and Arts-Rabigh, King Abdulaziz University, Rabigh 21911, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Damanhur University, Damanhur 22511, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Doaa A.","family":"Abdo","sequence":"additional","affiliation":[{"name":"Department of Applied Statistics and Insurance, Faculty of Commerce, Mansoura University, Mansoura 35511, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1080\/00031305.1995.10476150","article-title":"A simple simulation algorithm for generating progressively type-II censored samples","volume":"49","author":"Balakrishnan","year":"1995","journal-title":"Am. Stat."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1829","DOI":"10.1016\/j.spl.2012.06.007","article-title":"Objective Bayesian analysis of Pareto distribution under progressive Type-II censoring","volume":"82","author":"Fu","year":"2012","journal-title":"Stat. Probab. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1737","DOI":"10.1109\/TR.2016.2604298","article-title":"Generalized fiducial inference for accelerated life tests with Weibull distribution and progressively type-II censoring","volume":"65","author":"Chen","year":"2016","journal-title":"IEEE Trans. Reliab."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1109\/TR.2019.2948173","article-title":"A unified model for system reliability evaluation under dynamic operating conditions","volume":"70","author":"Xu","year":"2021","journal-title":"IEEE Trans. Reliab."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"108136","DOI":"10.1016\/j.ress.2021.108136","article-title":"Modelling and estimation of system reliability under dynamic operating environments and lifetime ordering constraints","volume":"218","author":"Luo","year":"2022","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"EL-Sagheer, R.M., Shokr, E.M., Mahmoud, M.A.W., and El-Desouky, B.S. (2021). Inferences for Weibull Fr\u00e9chet distribution using a Bayesian and Non-Bayesian methods on gastric cancer survival times. Comput. Math. Methods Med., 9965856.","DOI":"10.1155\/2021\/9965856"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1080\/02331880310001598864","article-title":"Estimation of the parameters of the Gompertz distribution under the first failure-censored sampling plan","volume":"37","author":"Wu","year":"2003","journal-title":"Statistics"},{"key":"ref_8","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":"10","author":"Wu","year":"2009","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Haj Ahmad, H., Salah, M.M., Eliwa, M.S., Ali Alhussain, Z., Almetwally, E.M., and Ahmed, E.A. (2021). Bayesian and non-Bayesian inference under adaptive type-II progressive censored sample with exponentiated power Lindley distribution. J. Appl. Stat.","DOI":"10.1080\/02664763.2021.1931819"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"161","DOI":"10.4236\/ojs.2011.13020","article-title":"Estimation of the unknown parameters for the compound Rayleigh distribution based on progressive first-failure-censored sampling","volume":"1","author":"Abushal","year":"2011","journal-title":"Open J. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2741","DOI":"10.1080\/02664763.2012.725466","article-title":"Estimation of the coefficient of variation for non-normal model using progressive first-failure-censoring data","volume":"12","author":"Soliman","year":"2012","journal-title":"J. Appl. Stat."},{"key":"ref_12","first-page":"117","article-title":"Estimation Based on Progressive First-Failure Censored Sampling with Binomial Removals","volume":"5","author":"Soliman","year":"2013","journal-title":"Intell. Inf. Manag."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"269","DOI":"10.12785\/jsap\/020310","article-title":"Bayesian Inference and Prediction using Progressive First-Failure Censored from Generalized Pareto Distribution","volume":"3","author":"Mahmoud","year":"2013","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2492","DOI":"10.1080\/02664763.2020.1815670","article-title":"Inference of progressively type-II censored competing risks data from Chen distribution with an application","volume":"47","author":"Ahmed","year":"2020","journal-title":"J. Appl. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Xie, Y., and Gui, W. (2020). Statistical inference of the lifetime performance index with the Log-Logistic distribution based on progressive first-failure-censored data. Symmetry, 12.","DOI":"10.3390\/sym12060937"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Shi, X., and Shi, Y. (2021). Inference for Inverse Power Lomax distribution with progressive first-failure censoring. Entropy, 23.","DOI":"10.3390\/e23091099"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2608","DOI":"10.1080\/02664763.2016.1142945","article-title":"The Weibull Frechet distribution and its applications","volume":"43","author":"Afify","year":"2016","journal-title":"J. Appl. Stat."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1007\/s00362-016-0787-2","article-title":"Estimation of parameters of Weibull\u2013Gamma distribution based on progressively censored data","volume":"59","year":"2018","journal-title":"Stat. Pap."},{"key":"ref_19","unstructured":"Greene, W.H. (2000). Econometric Analysis, Prentice-Hall. [4th ed.]."},{"key":"ref_20","unstructured":"Meeker, W.Q., and Escobar, L.A. (1998). Statistical Methods for Reliability Data, Wiley."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1214\/ss\/1032280214","article-title":"Bootstrap confidence intervals","volume":"11","author":"DiCiccio","year":"1996","journal-title":"Stat. Sci."},{"key":"ref_22","first-page":"927","article-title":"Theoretical comparison of bootstrap confidence intervals","volume":"16","author":"Hall","year":"1988","journal-title":"Ann. Stat."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Reiser, M., Yao, L., Wang, X., Wilcox, J., and Gray, S. (2017). A Comparison of Bootstrap Confidence Intervals for Multi-Level Longitudinal Data Using Monte-Carlo Simulation, in \u2018Monte-Carlo Simulation-Based Statistical Modeling\u2019, Springer.","DOI":"10.1007\/978-981-10-3307-0_17"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.cie.2018.12.037","article-title":"Evaluation of robust scale estimators for modified Weibull process capability indices and their bootstrap confidence intervals","volume":"128","author":"Besseris","year":"2019","journal-title":"Comput. Ind. Eng."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"EL-Sagheer, R.M., Eliwa, M.S., Alqahtani, K.M., and EL-Morshedy, M. (2022). Asymmetric randomly censored mortality distribution: Bayesian framework and parametric bootstrap with application to COVID-19 data. J. Math.","DOI":"10.1155\/2022\/8300753"},{"key":"ref_26","first-page":"1701","article-title":"Markov chains for exploring posterior distributions (with discussion)","volume":"22","author":"Tierney","year":"1994","journal-title":"Ann. Stat."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"104012","DOI":"10.1016\/j.rinp.2021.104012","article-title":"A new extended rayleigh distribution with applications of COVID-19 data","volume":"23","author":"Almongy","year":"2021","journal-title":"Results Phys."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/956\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:07:33Z","timestamp":1760137653000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/5\/956"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,7]]},"references-count":27,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["sym14050956"],"URL":"https:\/\/doi.org\/10.3390\/sym14050956","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,7]]}}}