{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:58:54Z","timestamp":1772254734673,"version":"3.50.1"},"reference-count":22,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2017,6,21]],"date-time":"2017-06-21T00:00:00Z","timestamp":1498003200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the program for the Fundamental Research Funds for the Central Universities","award":["2014RC042, 2015JBM109"],"award-info":[{"award-number":["2014RC042, 2015JBM109"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, we consider the problem of estimating stress-strength reliability for inverse Weibull lifetime models having the same shape parameters but different scale parameters. We obtain the maximum likelihood estimator and its asymptotic distribution. Since the classical estimator doesn\u2019t hold explicit forms, we propose an approximate maximum likelihood estimator. The asymptotic confidence interval and two bootstrap intervals are obtained. Using the Gibbs sampling technique, Bayesian estimator and the corresponding credible interval are obtained. The Metropolis-Hastings algorithm is used to generate random variates. Monte Carlo simulations are conducted to compare the proposed methods. Analysis of a real dataset is performed.<\/jats:p>","DOI":"10.3390\/a10020071","type":"journal-article","created":{"date-parts":[[2017,6,22]],"date-time":"2017-06-22T02:39:15Z","timestamp":1498099155000},"page":"71","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Bayesian and Classical Estimation of Stress-Strength Reliability for Inverse Weibull Lifetime Models"],"prefix":"10.3390","volume":"10","author":[{"given":"Qixuan","family":"Bi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China"}]},{"given":"Wenhao","family":"Gui","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China"}]}],"member":"1968","published-online":{"date-parts":[[2017,6,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Awad, A. (1985). Estimation of P(Y < X) in case of the double exponential distribution.","DOI":"10.1515\/9783112314036-064"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1007\/s13198-014-0267-9","article-title":"Estimation of P(Y < X) in Lindley distribution using progressively first failure censoring","volume":"6","author":"Kumar","year":"2016","journal-title":"Int. J. Syst. Assur. Eng. Manag."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1109\/TR.2006.874918","article-title":"Estimation of P[Y < X] for Weibull distributions","volume":"55","author":"Kundu","year":"2016","journal-title":"IEEE Trans. Reliab."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"480","DOI":"10.1016\/j.jspi.2009.07.024","article-title":"Estimation of P[Y < X] for generalized Pareto distribution","volume":"140","author":"Rezaei","year":"2010","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kizilaslan, F. (2016). Classical and Bayesian estimation of reliability in a multicomponent stress-strength model based on a general class of inverse exponentiated distributions. Stat. Pap., 1\u201332.","DOI":"10.1007\/s00362-016-0810-7"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"467","DOI":"10.15446\/rce.v38n2.51674","article-title":"Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution","volume":"38","author":"Kizilaslan","year":"2015","journal-title":"Rev. Colomb. Estad."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"190437","DOI":"10.1155\/2013\/190437","article-title":"On the Mean Residual Life Function and Stress and Strength Analysis under Different Loss Function for Lindley Distribution","volume":"2013","author":"Ali","year":"2013","journal-title":"J. Qual. Reliab. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.matcom.2016.10.011","article-title":"Classical and Bayesian estimation of reliability in a multicomponent stress-strength model based on the proportional reversed hazard rate mode","volume":"136","author":"Kizilaslan","year":"2017","journal-title":"Math. Comput. Simul."},{"key":"ref_9","first-page":"335","article-title":"Bayesian and non-bayesian estimation of stress\u2013strength model for Pareto type I distribution","volume":"37","author":"Shawky","year":"2013","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_10","unstructured":"Johnson, N.L., Kotz, S., and Balakrishnan, N. (1995). Distributions. Continuous Univariate Distributions, Wiley. [2nd ed.]."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1002\/qre.4680090426","article-title":"The complementary weibull distribution: Unknown or just forgotten?","volume":"9","author":"Drapella","year":"1993","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1016\/0143-8174(82)90036-1","article-title":"Reliability analysis of CNC machine tools","volume":"3","author":"Keller","year":"1982","journal-title":"Reliab. Eng."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.stamet.2009.11.001","article-title":"A discrete inverse Weibull distribution and estimation of its parameters","volume":"7","author":"Jazi","year":"2010","journal-title":"Stat. Methodol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5377","DOI":"10.1016\/j.csda.2006.09.016","article-title":"Mixture of two inverse Weibull distributions: Properties and estimation","volume":"51","author":"Sultan","year":"2007","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_15","first-page":"30","article-title":"Theoretical analysis of inverse weibull distribution","volume":"7","author":"Khan","year":"2008","journal-title":"Wseas Trans. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1007\/s00362-009-0271-3","article-title":"The generalized inverse Weibull distribution","volume":"52","author":"Gusmo","year":"2011","journal-title":"Stat. Pap."},{"key":"ref_17","first-page":"20","article-title":"Estimation and prediction for the inverse Weibull distribution based on records","volume":"2","author":"Riad","year":"2011","journal-title":"J. Adv. Res. Stat. Probab."},{"key":"ref_18","unstructured":"Arnold, B., and Balakrishnan, N. (2017, June 21). Relations, Bounds and Approximations for Order Statistics. Available online: http:\/\/www.springer.com\/gp\/book\/9780387969756."},{"key":"ref_19","unstructured":"Efron, B. (2017, June 21). The Jackknife, the Bootstrap and Other Resampling Plans. Available online: http:\/\/epubs.siam.org\/doi\/book\/10.1137\/1.9781611970319."},{"key":"ref_20","first-page":"927","article-title":"Rejoinder: Theoretical comparison of bootstrap confidence intervals","volume":"16","author":"Hall","year":"1988","journal-title":"Ann. Stat."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1080\/10618600.1999.10474802","article-title":"Monte Carlo estimation of Bayesian credible and HPD intervals","volume":"8","author":"Chen","year":"1999","journal-title":"J. Comput. Graph. Stat."},{"key":"ref_22","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."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/10\/2\/71\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T18:39:51Z","timestamp":1760207991000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/10\/2\/71"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,21]]},"references-count":22,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,6]]}},"alternative-id":["a10020071"],"URL":"https:\/\/doi.org\/10.3390\/a10020071","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,6,21]]}}}