{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:54:11Z","timestamp":1760237651112,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,6,3]],"date-time":"2020-06-03T00:00:00Z","timestamp":1591142400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Estimating the accurate evaluation of product lifetime performance has always been a hot topic in manufacturing industry. This paper, based on the lifetime performance index, focuses on its evaluation when a lower specification limit is given. The progressive first-failure-censored data we discuss have a common log-logistic distribution. Both Bayesian and non-Bayesian method are studied. Bayes estimator of the parameters of the log-logistic distribution and the lifetime performance index are obtained using both the Lindley approximation and Monte Carlo Markov Chain methods under symmetric and asymmetric loss functions. As for interval estimation, we apply the maximum likelihood estimator to construct the asymptotic confidence intervals and the Metropolis\u2013Hastings algorithm to establish the highest posterior density credible intervals. Moreover, we analyze a real data set for demonstrative purposes. In addition, different criteria for deciding the optimal censoring scheme have been studied.<\/jats:p>","DOI":"10.3390\/sym12060937","type":"journal-article","created":{"date-parts":[[2020,6,5]],"date-time":"2020-06-05T03:32:21Z","timestamp":1591327941000},"page":"937","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Statistical Inference of the Lifetime Performance Index with the Log-Logistic Distribution Based on Progressive First-Failure-Censored Data"],"prefix":"10.3390","volume":"12","author":[{"given":"Ying","family":"Xie","sequence":"first","affiliation":[{"name":"Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4318-1780","authenticated-orcid":false,"given":"Wenhao","family":"Gui","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,6,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Cooper, T. (2016). Longer Lasting Products: Alternatives to the Throwaway Society, CRC Press.","DOI":"10.4324\/9781315592930"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.resconrec.2019.104437","article-title":"Advancing quantitative rigor in the circular economy literature: New methodology for product lifetime extension business models","volume":"150","author":"Ertz","year":"2019","journal-title":"Resour. Conserv. Recycl."},{"key":"ref_3","unstructured":"Montgomery, D.C. (1991). Introduction to Statistical Quality Control, John Wiley & Sons."},{"key":"ref_4","first-page":"116","article-title":"Computational testing algorithmic procedure of assessment for lifetime performance index of products with two-parameter exponential distribution","volume":"190","author":"Wu","year":"2007","journal-title":"Appl. Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Lee, W.C., Wu, J.W., and Hong, C.W. (2009). Assessing the Lifetime Performance Index of Products with the Exponential Distribution under Progressively Type II Right Censored Samples, Elsevier Science.","DOI":"10.1016\/j.cam.2009.04.018"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1305","DOI":"10.1080\/00207721.2010.494774","article-title":"Implementing lifetime performance index of products with two-parameter exponential distribution","volume":"42","author":"Lee","year":"2011","journal-title":"Int. J. Syst. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2248","DOI":"10.1080\/00949655.2013.788652","article-title":"Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive type-II censoring","volume":"84","author":"Sultan","year":"2014","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1080\/02664763.2016.1183602","article-title":"Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring","volume":"44","author":"Lee","year":"2016","journal-title":"J. Appl. Stat."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1080\/00401706.1963.10490102","article-title":"Progressively Censored Samples in Life Testing","volume":"5","author":"Cohen","year":"1963","journal-title":"Technometrics"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1016\/j.joems.2017.06.001","article-title":"Statistical inferences based on an adaptive progressive type-II censoring from exponentiated exponential distribution","volume":"25","author":"Mohammed","year":"2017","journal-title":"J. Egypt. Math. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Balakrishnan, N., and Aggarwala, R. (2000). Progressive Censoring: Theory, Methods, and Applications, Birkh\u00e4user.","DOI":"10.1007\/978-1-4612-1334-5"},{"key":"ref_12","unstructured":"Johnson, L.G. (1964). Theory And Technique Of Variation Research, Elsevier Pub. Co."},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"1576","DOI":"10.1080\/02664763.2016.1214692","article-title":"Estimation and prediction for the generalized inverted exponential distribution based on progressively first-failure-censored data with application","volume":"44","author":"Ahmed","year":"2017","journal-title":"J. Appl. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/s00180-015-0622-6","article-title":"On progressively first failure censored Lindley distribution","volume":"31","author":"Madhulika","year":"2016","journal-title":"Comput. Stat."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2471","DOI":"10.1016\/j.csda.2012.01.025","article-title":"Estimation of the parameters of life for Gompertz distribution using progressive first-failure censored data","volume":"56","author":"Soliman","year":"2012","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.cam.2012.09.006","article-title":"Estimating the lifetime performance index with Weibull distribution based on progressive first-failure censoring scheme","volume":"239","author":"Ahmadi","year":"2013","journal-title":"J. Comput. Appl. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1080\/00949655.2015.1052440","article-title":"Generalized inverted exponential distribution under progressive first-failure censoring","volume":"86","author":"Madhulika","year":"2016","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1063\/1.4907435","article-title":"Bayesian MCMC inference for the Gompertz distribution based on progressive first-failure censoring data","volume":"1643","author":"Ahmed","year":"2015","journal-title":"AIP Conf. Proc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1196","DOI":"10.1080\/02664763.2018.1541170","article-title":"Pareto analysis for the lifetime performance index of products on the basis of progressively first-failure-censored batches under balanced symmetric and asymmetric loss functions","volume":"46","author":"Ahmadi","year":"2019","journal-title":"J. Appl. Stat."},{"key":"ref_21","first-page":"162","article-title":"Log-logistic Distribution as a Reliability Model: A Bayesian Analysis","volume":"4","author":"Akhtar","year":"2014","journal-title":"Am. J. Math. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1774","DOI":"10.1186\/s40064-016-3476-7","article-title":"Log-logistic distribution for survival data analysis using MCMC","volume":"5","author":"Shawky","year":"2016","journal-title":"Springerplus"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1007\/s00184-012-0397-5","article-title":"Estimation of the scale parameter of a log-logistic distribution","volume":"76","author":"Lesitha","year":"2013","journal-title":"Metrika"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1080\/00401706.1998.10485526","article-title":"Escobar Statistical Methods for Reliability Data","volume":"40","author":"William","year":"1998","journal-title":"Technometrics"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/BF02888353","article-title":"Approximate Bayesian methods","volume":"31","author":"Lindley","year":"1980","journal-title":"Trab. Estad. Y Investig. Oper."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s41096-018-0032-5","article-title":"Analysis of Weibull Distribution Under Adaptive Type-II Progressive Hybrid Censoring Scheme","volume":"19","author":"Nassar","year":"2018","journal-title":"J. Ind. Soc. Probab. Stat."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s11222-008-9069-8","article-title":"Bayesian computation with R","volume":"19","author":"Albert","year":"2009","journal-title":"Stat. Comput."},{"key":"ref_28","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_29","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1002\/qre.691","article-title":"A Bootstrap Control Chart for Weibull Percentiles","volume":"22","author":"Nichols","year":"2006","journal-title":"Qual. Reliab. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1479","DOI":"10.1080\/02664763.2016.1214245","article-title":"Estimation based on progressive first-failure censoring from exponentiated exponential distribution","volume":"44","author":"Mohammed","year":"2017","journal-title":"J. Appl. Stat."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1007\/s11749-008-0110-1","article-title":"On progressively censored generalized exponential distribution","volume":"18","author":"Pradhan","year":"2009","journal-title":"Test"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1111\/jiec.12610","article-title":"Product Design in a Circular Economy: Development of a Typology of Key Concepts and Terms","volume":"21","author":"Bakker","year":"2017","journal-title":"J. Ind. Ecol."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1016\/j.jclepro.2019.06.264","article-title":"Made to break? A taxonomy of business models on product lifetime extension","volume":"234","author":"Ertz","year":"2019","journal-title":"J. Clean. 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