{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T18:32:40Z","timestamp":1770748360883,"version":"3.49.0"},"reference-count":23,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,22]],"date-time":"2021-07-22T00:00:00Z","timestamp":1626912000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004663","name":"Ministry of Science and Technology, Taiwan","doi-asserted-by":"publisher","award":["MOST 109-2118- M-032-001-MY2"],"award-info":[{"award-number":["MOST 109-2118- M-032-001-MY2"]}],"id":[{"id":"10.13039\/501100004663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004663","name":"Ministry of Science and Technology, Taiwan","doi-asserted-by":"publisher","award":["MOST 108- 2118-M-032-001-"],"award-info":[{"award-number":["MOST 108- 2118-M-032-001-"]}],"id":[{"id":"10.13039\/501100004663","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>With the high demands on the quality of high-tech products for consumers, assuring the lifetime performance is a very important task for competitive manufacturing industries. The lifetime performance index CL is frequently used to monitor the larger-the-better lifetime performance of products. This research is related to the topic of asymmetrical probability distributions and applications across disciplines. Chen lifetime distribution with a bathtub shape or increasing failure rate function has many applications in the lifetime data analysis. We derived the uniformly minimum variance unbiased estimator (UMVUE) for CL, and we used this estimator to develop a hypothesis testing procedure of CL under a lower specification limit based on the progressive type-II censored sample. The Bayesian estimator for CL is also derived, and it is used to develop another hypothesis testing procedure. A simulation study is conducted to compare the average confidence levels for two procedures. Finally, one practical example is given to illustrate the implementation of our proposed non-Bayesian and Bayesian testing procedure.<\/jats:p>","DOI":"10.3390\/sym13081322","type":"journal-article","created":{"date-parts":[[2021,7,22]],"date-time":"2021-07-22T22:37:14Z","timestamp":1626993434000},"page":"1322","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Bayesian Testing Procedure on the Lifetime Performance Index of Products Following Chen Lifetime Distribution Based on the Progressive Type-II Censored Sample"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0611-6701","authenticated-orcid":false,"given":"Shu-Fei","family":"Wu","sequence":"first","affiliation":[{"name":"Department of Statistics, Tamkang University, Tamsui, Taipei 251301, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei-Tsung","family":"Chang","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Taipei, Taipei 100234, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,22]]},"reference":[{"key":"ref_1","unstructured":"Montgomery, D.C. (1985). Introduction to Statistical Quality Control, John Wiley and Sons."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"812","DOI":"10.1108\/02656710210434757","article-title":"Statistical testing for assessing the performance of lifetime index of electronic components with exponential distribution","volume":"19","author":"Tong","year":"2002","journal-title":"Int. J. Qual. Reliab. Manag."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Balakrishnan, N., and Cramer, E. (2014). The Art of Progressive Censoring. Applications to Reliability and Quality, Birkh\u00e4user.","DOI":"10.1007\/978-0-8176-4807-7"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1921","DOI":"10.1081\/STA-100105705","article-title":"Progressive interval censoring: Some mathematical results with applications to inference","volume":"30","author":"Aggarwala","year":"2001","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1007\/s11135-008-9187-6","article-title":"Interval estimation for the two-parameter exponential distribution under progressive censoring","volume":"44","author":"Wu","year":"2010","journal-title":"Qual. Quant."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1080\/00949650802110433","article-title":"The Exact Hypothesis Test for the Shape Parameter of a New Two-Parameter Distribution with the Bathtub Shape or Increasing Failure Rate function under Progressive Censoring with Random Removals","volume":"79","author":"Wu","year":"2009","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.cam.2017.11.022","article-title":"The performance assessment on the lifetime performance index of products following Chen lifetime distribution based on the progressive type I interval censored sample","volume":"334","author":"Wu","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Wu, S.F., and Chang, W.T. (2021). The evaluation on the process capability index CL for exponentiated Frech\u2019et lifetime product under progressive type I interval censoring. Symmetry, 13.","DOI":"10.3390\/sym13061032"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"66","DOI":"10.1016\/j.cam.2018.10.044","article-title":"The assessment on the lifetime performance index of products with Gompertz distribution based on the progressive type I interval censored sample","volume":"351","author":"Wu","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1515\/eqc-2015-0008","article-title":"Inference for the lifetime performance index with progressively Type-II censored samples from gamma distributions","volume":"30","author":"Laumen","year":"2015","journal-title":"Econ. Qual. Control"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"648","DOI":"10.1016\/j.cam.2009.04.018","article-title":"Assessing the lifetime performance index of products with the exponential distribution under progressively type II right censored samples","volume":"231","author":"Lee","year":"2009","journal-title":"J. Comput. Appl. Math."},{"key":"ref_12","first-page":"57","article-title":"Bayesian test of lifetime performance index for exponential products based on the progressively type II right censored sample","volume":"8","author":"Wu","year":"2011","journal-title":"J. Quant. Manag."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2167","DOI":"10.1016\/j.matcom.2008.12.001","article-title":"Assessing the lifetime performance index of products from progressively type II right censored data using Burr XII model","volume":"79","author":"Lee","year":"2009","journal-title":"Math. Comput. Simul."},{"key":"ref_14","first-page":"671","article-title":"Implementing Lifetime Performance Index of Burr XII Products with Progressively Type II Right Censored Sample","volume":"10","author":"Wu","year":"2014","journal-title":"Int. J. Innov. Comput. Inf. Control"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1676","DOI":"10.1016\/j.cam.2010.09.009","article-title":"Assessing the lifetime performance index of Rayleigh products based on the bayesian estimation under progressive type II right censored samples","volume":"235","author":"Lee","year":"2011","journal-title":"J. Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1016\/S0167-7152(00)00044-4","article-title":"A new two-parameter lifetime distribution with bathtub shape or increasing failure rate function","volume":"49","author":"Chen","year":"2000","journal-title":"Stat. Probab. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.ress.2014.04.026","article-title":"A new class of distribution functions for lifetime data","volume":"129","author":"Domma","year":"2014","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Balakrishnan, N., and Aggarwala, R. (2000). Progressive Censoring: Theory, Methods and Applications, Birkhauser.","DOI":"10.1007\/978-1-4612-1334-5"},{"key":"ref_19","unstructured":"Fussell, J.B., and Burdick, G.R. (1977). Gamma prior distribution selection for Bayesian analysis of failure rate and reliability. Nuclear Systems Reliability Engineering and Risk Assessment, SIAM."},{"key":"ref_20","unstructured":"Casella, G., and Berger, R.L. (2002). Statistical Inference, Duxbury Press. [2nd ed.]."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/0951-8320(95)00149-2","article-title":"Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function","volume":"52","author":"Xie","year":"1995","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1111\/j.2517-6161.1978.tb01048.x","article-title":"A scale-free goodness-of-fit Test for the Exponential Distribution Based on the Gini Statistic","volume":"40","author":"Gill","year":"1978","journal-title":"J. R. Stat. Soc."},{"key":"ref_23","first-page":"305","article-title":"Completeness, Similar Regions and Unbiased Estimates","volume":"10","author":"Lehmann","year":"1950","journal-title":"Sankhya"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1322\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:33:37Z","timestamp":1760164417000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/8\/1322"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,22]]},"references-count":23,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["sym13081322"],"URL":"https:\/\/doi.org\/10.3390\/sym13081322","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,22]]}}}