{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T03:27:39Z","timestamp":1781753259510,"version":"3.54.5"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2024,11,14]],"date-time":"2024-11-14T00:00:00Z","timestamp":1731542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,11,14]],"date-time":"2024-11-14T00:00:00Z","timestamp":1731542400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2024,12]]},"DOI":"10.1007\/s11222-024-10534-1","type":"journal-article","created":{"date-parts":[[2024,11,14]],"date-time":"2024-11-14T11:40:25Z","timestamp":1731584425000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A novel approach for parameter estimation of mixture of two Weibull distributions in failure data modeling"],"prefix":"10.1007","volume":"34","author":[{"given":"Tianyu","family":"Yan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kai-Tai","family":"Fang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hong","family":"Yin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,11,14]]},"reference":[{"issue":"2","key":"10534_CR1","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1016\/j.ress.2004.05.003","volume":"87","author":"L Attardi","year":"2005","unstructured":"Attardi, L., Guida, M., Pulcini, G.: A mixed-weibull regression model for the analysis of automotive warranty data. Reliabil Eng Syst Safety 87(2), 265\u2013273 (2005)","journal-title":"Reliabil Eng Syst Safety"},{"issue":"1","key":"10534_CR2","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/0015-0568(82)90015-X","volume":"16","author":"CP Beetz Jr","year":"1982","unstructured":"Beetz, C.P., Jr.: The analysis of carbon fibre strength distributions exhibiting multiple modes of failure. Fibre Sci Technol 16(1), 45\u201359 (1982)","journal-title":"Fibre Sci Technol"},{"issue":"4","key":"10534_CR3","doi-asserted-by":"publisher","first-page":"377","DOI":"10.1109\/TR.1982.5221382","volume":"31","author":"SW Cheng","year":"1982","unstructured":"Cheng, S.W., Fu, J.C.: Estimation of mixed weibull parameters in life testing. IEEE Trans Reliabil 31(4), 377\u2013381 (1982)","journal-title":"IEEE Trans Reliabil"},{"key":"10534_CR4","volume-title":"Survival Models and Data Analysis. Wiley Series in Probability and Mathematical Statistics","author":"R Eldandt-Johnson","year":"1980","unstructured":"Eldandt-Johnson, R., Johnson, N.: Survival Models and Data Analysis. Wiley Series in Probability and Mathematical Statistics. Wiley, Indianapolis (1980)"},{"key":"10534_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-5897-5","volume-title":"Finite Mixture Distributions","author":"BS Everitt","year":"1981","unstructured":"Everitt, B.S., Hand, D.J.: Finite Mixture Distributions. Springer, Netherlands (1981)"},{"issue":"2","key":"10534_CR6","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1080\/00401706.1970.10488679","volume":"12","author":"LW Falls","year":"1970","unstructured":"Falls, L.W.: Estimation of parameters in compound weibull distributions. Technometrics 12(2), 399\u2013407 (1970)","journal-title":"Technometrics"},{"key":"10534_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/978-981-13-2041-5","volume-title":"Theory and Application of Uniform Experimental Designs","author":"KT Fang","year":"2018","unstructured":"Fang, K.T., Liu, M., Qin, H., et al.: Theory and Application of Uniform Experimental Designs. Springer, Singapore (2018)"},{"issue":"13","key":"10534_CR8","doi-asserted-by":"publisher","first-page":"2930","DOI":"10.3390\/math11132930","volume":"11","author":"KT Fang","year":"2023","unstructured":"Fang, K.T., Pan, J.: A review of representative points of statistical distributions and their applications. Mathematics 11(13), 2930 (2023)","journal-title":"Mathematics"},{"key":"10534_CR9","unstructured":"Fang, K. T., Wang, Y.: A sequential algorithm for solving a system of nonlinear equations. J. Comput. Math., pp. 9\u201316 (1991)"},{"key":"10534_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-3095-8","volume-title":"Number-Theoretic Methods in Statistics","author":"KT Fang","year":"1994","unstructured":"Fang, K.T., Wang, Y.: Number-Theoretic Methods in Statistics, 1st edn. Chapman and Hall, London (1994)","edition":"1"},{"issue":"3","key":"10534_CR11","doi-asserted-by":"publisher","first-page":"416","DOI":"10.1214\/ss\/1177010392","volume":"9","author":"KT Fang","year":"1994","unstructured":"Fang, K.T., Wang, Y., Bentler, P.M.: Some applications of number-theoretic methods in statistics. Stat Sci 9(3), 416\u2013428 (1994)","journal-title":"Stat Sci"},{"key":"10534_CR12","doi-asserted-by":"crossref","unstructured":"Fisher, R. A.: On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. Lond. Ser. A Contain. Pap. Math. Phys. Char. 222, 309\u2013368 (1922)","DOI":"10.1098\/rsta.1922.0009"},{"issue":"3","key":"10534_CR13","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/0167-7152(89)90135-1","volume":"8","author":"JC Fu","year":"1989","unstructured":"Fu, J.C.: Method of Kim-zam: an algorithm for computing the maximum likelihood estimator. Stat. Prob. Lett. 8(3), 289\u2013296 (1989)","journal-title":"Stat. Prob. Lett."},{"issue":"3","key":"10534_CR14","doi-asserted-by":"publisher","first-page":"635","DOI":"10.1093\/biomet\/69.3.635","volume":"69","author":"FE Harrell","year":"1982","unstructured":"Harrell, F.E., Davis, C.: A new distribution-free quantile estimator. Biometrika 69(3), 635\u2013640 (1982)","journal-title":"Biometrika"},{"issue":"3","key":"10534_CR15","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1109\/24.406588","volume":"44","author":"R Jiang","year":"1995","unstructured":"Jiang, R., Murthy, D.: Modeling failure-data by mixture of 2 weibull distributions: a graphical approach. IEEE Trans. Reliabil. 44(3), 477\u2013488 (1995)","journal-title":"IEEE Trans. Reliabil."},{"issue":"2","key":"10534_CR16","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1109\/24.257789","volume":"41","author":"S Jiang","year":"1992","unstructured":"Jiang, S., Kececioglu, D.: Graphical representation of two mixed-weibull distributions. IEEE Trans. Reliabil. 41(2), 241\u2013247 (1992)","journal-title":"IEEE Trans. Reliabil."},{"issue":"2","key":"10534_CR17","doi-asserted-by":"publisher","first-page":"248","DOI":"10.1109\/24.257791","volume":"41","author":"S Jiang","year":"1992","unstructured":"Jiang, S., Kececioglu, D.: Maximum likelihood estimates, from censored data, for mixed-weibull distributions. IEEE Trans. Reliabil. 41(2), 248\u2013255 (1992)","journal-title":"IEEE Trans. Reliabil."},{"key":"10534_CR18","unstructured":"Johnson, L. G.: The Statistical Treatment of Fatigue Experiments. ELSEVIER (1964)"},{"issue":"19","key":"10534_CR19","doi-asserted-by":"publisher","first-page":"2217","DOI":"10.1080\/03610926208828383","volume":"11","author":"W Kalgh","year":"1982","unstructured":"Kalgh, W., Lachenbruch, P.A.: A generalized quantile estimator. Commun. Stat. Theory Methods 11(19), 2217\u20132238 (1982)","journal-title":"Commun. Stat. Theory Methods"},{"key":"10534_CR20","doi-asserted-by":"crossref","unstructured":"Kececioglu, D., Wang, W.: Parameter estimation for mixed-weibull distribution. In: Proceedings of the Reliability and Maintainability Symposium, Anaheim, CA, USA, pp. 247\u2013252, 19\u201322 January (1998)","DOI":"10.1109\/RAMS.1998.653782"},{"key":"10534_CR21","volume-title":"Statistical Models and Methods for Lifetime Data","author":"JF Lawless","year":"2011","unstructured":"Lawless, J.F.: Statistical Models and Methods for Lifetime Data. John Wiley & Sons, New York (2011)"},{"key":"10534_CR22","unstructured":"Li, Y., Fang, K. T.: A new approach to parameter estimation of mixture of two normal distributions. Commun. Stat. Simul. Comput., pp. 1\u201327 (2022)"},{"key":"10534_CR23","doi-asserted-by":"crossref","unstructured":"Ling, D., Huang, H., Liu, Y.: A method for parameter estimation of mixed weibull distribution. In: Proceedings of Annual Reliability and Maintainability Symposium, pp. 129\u2013133 (2009)","DOI":"10.1109\/RAMS.2009.4914663"},{"issue":"3","key":"10534_CR24","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1016\/S0951-8320(97)00171-3","volume":"60","author":"J Ling","year":"1998","unstructured":"Ling, J., Pan, J.: A new method for selection of population distribution and parameter estimation. Reliabil. Eng. Syst. Safety 60(3), 247\u2013255 (1998)","journal-title":"Reliabil. Eng. Syst. Safety"},{"key":"10534_CR25","unstructured":"Mann, N., Schaefer, R., Singpurwalla, N.: Methods for Statistical Analysis of Reliability and Life Data. John Wiley and Sons, New York (1974)"},{"issue":"3\u20134","key":"10534_CR26","doi-asserted-by":"publisher","first-page":"504","DOI":"10.1093\/biomet\/45.3-4.504","volume":"45","author":"W Mendenhall","year":"1958","unstructured":"Mendenhall, W., Hader, R.: Estimation of parameters of mixed exponentially distributed failure time distributions from censored life test data. Biometrika 45(3\u20134), 504\u2013520 (1958)","journal-title":"Biometrika"},{"issue":"3","key":"10534_CR27","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1111\/bmsp.12198","volume":"73","author":"G Navruz","year":"2020","unstructured":"Navruz, G., \u00d6zdemir, A.F.: A new quantile estimator with weights based on a subsampling approach. Br. J. Math. Stat. Psychol. 73(3), 50\u2013521 (2020)","journal-title":"Br. J. Math. Stat. Psychol."},{"key":"10534_CR28","doi-asserted-by":"publisher","DOI":"10.1002\/0471725234","volume-title":"Applied Life Data Analysis","author":"W Nelson","year":"1982","unstructured":"Nelson, W.: Applied Life Data Analysis. Wiley, New York (1982)"},{"key":"10534_CR29","doi-asserted-by":"crossref","unstructured":"Niederreiter, H.: Random number generation and Quasi-Monte Carlo methods. Soc. Industr. Appl. Math. (SIAM), Philadelphia (1992)","DOI":"10.1137\/1.9781611970081"},{"issue":"2","key":"10534_CR30","doi-asserted-by":"publisher","first-page":"660","DOI":"10.1137\/0907044","volume":"7","author":"H Niederreiter","year":"1986","unstructured":"Niederreiter, H., Peart, P.: Localization of search in Quasi-Monte Carlo methods for global optimization. SIAM J. Sci. Stat. Comput. 7(2), 660\u2013664 (1986)","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"10534_CR31","doi-asserted-by":"publisher","DOI":"10.1201\/9781420087444","volume-title":"The Weibull Distribution: A Handbook","author":"H Rinne","year":"2008","unstructured":"Rinne, H.: The Weibull Distribution: A Handbook. Chapman and Hall\/CRC, London (2008)"},{"issue":"2","key":"10534_CR32","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1080\/03610910701790491","volume":"37","author":"ME Sfakianakis","year":"2008","unstructured":"Sfakianakis, M.E., Verginis, D.G.: A new family of nonparametric quantile estimators. Commun. Stat. Simul. Comput. 37(2), 337\u2013345 (2008)","journal-title":"Commun. Stat. Simul. Comput."},{"key":"10534_CR33","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2021.104829","volume":"188","author":"J Yang","year":"2021","unstructured":"Yang, J., He, P., Fang, K.T.: Three kinds of discrete approximations of statistical multivariate distributions and their applications. J. Multivariate Anal. 188, 104829 (2021)","journal-title":"J. Multivariate Anal."},{"issue":"392","key":"10534_CR34","doi-asserted-by":"publisher","first-page":"1004","DOI":"10.1080\/01621459.1985.10478217","volume":"80","author":"SS Yang","year":"1985","unstructured":"Yang, S.S.: A smooth nonparametric estimator of a quantile function. J. Am. Stat. Assoc. 80(392), 1004\u20131011 (1985)","journal-title":"J. Am. Stat. Assoc."},{"key":"10534_CR35","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1016\/j.ress.2013.02.004","volume":"115","author":"T Zhang","year":"2013","unstructured":"Zhang, T., Dwight, R.: Choosing an optimal model for failure data analysis by graphical approach. Reliabil. Eng. Syst. Safety 115, 111\u2013123 (2013)","journal-title":"Reliabil. Eng. Syst. Safety"}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-024-10534-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-024-10534-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-024-10534-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T07:02:45Z","timestamp":1732518165000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-024-10534-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,14]]},"references-count":35,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["10534"],"URL":"https:\/\/doi.org\/10.1007\/s11222-024-10534-1","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-4265084\/v1","asserted-by":"object"}]},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,14]]},"assertion":[{"value":"14 April 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 November 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 November 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"221"}}