{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:53:50Z","timestamp":1760234030660,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,16]],"date-time":"2021-03-16T00:00:00Z","timestamp":1615852800000},"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>Point and interval estimations are taken into account for a progressive first-failure censored left-truncated normal distribution in this paper. First, we derive the estimators for parameters on account of the maximum likelihood principle. Subsequently, we construct the asymptotic confidence intervals based on these estimates and the log-transformed estimates using the asymptotic normality of maximum likelihood estimators. Meanwhile, bootstrap methods are also proposed for the construction of confidence intervals. As for Bayesian estimation, we implement the Lindley approximation method to determine the Bayesian estimates under not only symmetric loss function but also asymmetric loss functions. The importance sampling procedure is applied at the same time, and the highest posterior density (HPD) credible intervals are established in this procedure. The efficiencies of classical statistical and Bayesian inference methods are evaluated through numerous simulations. We conclude that the Bayes estimates given by Lindley approximation under Linex loss function are highly recommended and HPD interval possesses the narrowest interval length among the proposed intervals. Ultimately, we introduce an authentic dataset describing the tensile strength of 50mm carbon fibers as an illustrative sample.<\/jats:p>","DOI":"10.3390\/sym13030490","type":"journal-article","created":{"date-parts":[[2021,3,16]],"date-time":"2021-03-16T21:42:41Z","timestamp":1615930961000},"page":"490","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Classical and Bayesian Inference for a Progressive First-Failure Censored Left-Truncated Normal Distribution"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3563-7023","authenticated-orcid":false,"given":"Yuxin","family":"Cai","sequence":"first","affiliation":[{"name":"Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China"}]},{"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"}]}],"member":"1968","published-online":{"date-parts":[[2021,3,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Balakrishnan, N., and Aggarwala, R. 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Stat."},{"key":"ref_5","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_6","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":"Volume 1643","author":"Soliman","year":"2015","journal-title":"AIP Conference Proceedings"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s42519-019-0052-9","article-title":"Inference for the Chen distribution under progressive first-failure censoring","volume":"13","author":"Kayal","year":"2019","journal-title":"J. Stat. Theory Pract."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"6662","DOI":"10.1166\/jctn.2016.5612","article-title":"MCMC in analysis of progressively first failure censored competing risks data for gompertz model","volume":"13","author":"Bakoban","year":"2016","journal-title":"J. Comput. Theor. Nanosci."},{"key":"ref_9","first-page":"403","article-title":"Estimation in step-stress accelerated life tests for Weibull distribution with progressive first-failure censoring","volume":"3","author":"Aly","year":"2015","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_10","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_11","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1080\/03610918708812582","article-title":"Estimating the parameters of a doubly truncated normal distribution","volume":"16","author":"Mittal","year":"1987","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"357","DOI":"10.1080\/00031305.1999.10474490","article-title":"Mean and variance of truncated normal distributions","volume":"53","author":"Barr","year":"1999","journal-title":"Am. Stat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"847","DOI":"10.1080\/03610920701693876","article-title":"Sample size and power calculations for left-truncated normal distribution","volume":"37","author":"Ren","year":"2008","journal-title":"Commun. Stat. Methods"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1402","DOI":"10.1080\/02664763.2019.1679096","article-title":"Inference on a progressive type I interval-censored truncated normal distribution","volume":"47","author":"Lodhi","year":"2020","journal-title":"J. Appl. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1504\/IJEDPO.2013.059667","article-title":"Rethinking the truncated normal distribution","volume":"3","author":"Cha","year":"2013","journal-title":"Int. J. Exp. Des. Process Optim."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1260","DOI":"10.1016\/j.renene.2017.09.043","article-title":"Estimation of wind speed probability density function using a mixture of two truncated normal distributions","volume":"115","author":"Mazzeo","year":"2018","journal-title":"Renew. Energy"},{"key":"ref_17","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":"Dube","year":"2016","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_18","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_19","doi-asserted-by":"crossref","unstructured":"Efron, B., and Tibshirani, R.J. (1993). An Introduction to the Bootstrap, Chapman and Hall Inc.","DOI":"10.1007\/978-1-4899-4541-9"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"446","DOI":"10.1080\/01621459.1986.10478289","article-title":"Bayesian estimation and prediction using asymmetric loss functions","volume":"81","author":"Zellner","year":"1986","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_21","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. De Estad\u00edstica Y De Investig. Oper."},{"key":"ref_22","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_23","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. Int."},{"key":"ref_24","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. 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