{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:20:36Z","timestamp":1760145636142,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,16]],"date-time":"2024-08-16T00:00:00Z","timestamp":1723766400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science and Technology Council, Taiwan","award":["NSTC 113-2118-M-032-002"],"award-info":[{"award-number":["NSTC 113-2118-M-032-002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The lifetime performance index is commonly used in the manufacturing industry to evaluate the performance of the capabilities of the production process. For products with multiple components, the comprehensive lifetime performance index, which is a monotonically increasing function of the overall process yield, is used to relate to each individual lifetime performance index. For products where the lifetime of the ith component follows an exponentiated Fr\u00e9chet lifetime distribution, we examine the maximum likelihood estimators for both the comprehensive and individual lifetime performance indices based on the progressive type I interval-censored samples, deriving their asymptotic distributions. By specifying the target level for the comprehensive lifetime performance index, we can set the desired level for individual indices. A testing procedure, using the maximum likelihood estimator as the test statistic, was developed to determine if the comprehensive lifetime performance index meets the target. Given that the lifetime distribution is asymmetric, this study pertains to asymmetrical probability distributions and their applications across diverse fields. We illustrate the power analysis of this testing procedure with figures and summarize key findings. Finally, we demonstrate the application of this testing algorithm with a practical example involving two components to verify if the overall production process achieves the assigned target level.<\/jats:p>","DOI":"10.3390\/sym16081060","type":"journal-article","created":{"date-parts":[[2024,8,16]],"date-time":"2024-08-16T09:15:41Z","timestamp":1723799741000},"page":"1060","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Computational Analysis of the Comprehensive Lifetime Performance Index for Exponentiated Fr\u00e9chet Lifetime Distribution Products with Multi-Components"],"prefix":"10.3390","volume":"16","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, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hsueh-Chien","family":"Yeh","sequence":"additional","affiliation":[{"name":"Department of Statistics, Tamkang University, Tamsui, Taipei 251301, Taiwan, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,16]]},"reference":[{"key":"ref_1","unstructured":"Juran, J.M. (1974). Juran\u2019s Quality Control Handbook, McGraw-Hill. [3rd ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1080\/00224065.1986.11978984","article-title":"Process capability indices","volume":"18","author":"Kane","year":"1986","journal-title":"J. Qual. Technol."},{"key":"ref_3","unstructured":"Hsiang, T.C., and Taguchi, G. (1985). A tutorial on quality control and assurance\u2014The Taguchi methods. Joint Meetings of the American Statistical Association, InASA Annual Meeting LA."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1080\/00224065.1992.11979403","article-title":"Distributional and inferential properties of process capability indices","volume":"24","author":"Pearn","year":"1992","journal-title":"J. Qual. Technol."},{"key":"ref_5","unstructured":"Montgomery, D.C. (1985). Introduction to Statistical Quality Control, John Wiley and Sons Inc."},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/s11749-007-0061-y","article-title":"Progressive censoring methodology: An appraisal","volume":"16","author":"Balakrishnan","year":"2007","journal-title":"Test"},{"key":"ref_8","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_9","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_10","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_11","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1080\/00949650701216612","article-title":"A Laguerre polynomial approximation for a goodness-of-fit test for exponential distribution based on progressively censored data","volume":"78","author":"Sanjel","year":"2008","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1016\/j.cam.2016.08.005","article-title":"Computational testing algorithmic procedure of assessment for lifetime performance index of products with weibull distribution under progressive type I interval censoring","volume":"311","author":"Wu","year":"2017","journal-title":"J. Comput. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2670","DOI":"10.1080\/03610918.2017.1359282","article-title":"A hypothesis testing procedure for the evaluation on the lifetime performance index of products with Burr XII distribution under progressive type I interval censoring","volume":"47","author":"Wu","year":"2018","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_15","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_16","doi-asserted-by":"crossref","unstructured":"Wu, S.F., and Chiang, K.Y. (2024). Assessment of the overall lifetime performance index of Weibull products in multiple production lines. Mathematics, 12.","DOI":"10.3390\/math12040514"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Wu, S.F., Kuo, P.H., and Deng, W.S. (2024). The computational testing procedure for the comprehensive lifetime performance index of Burr XII products in multiple production lines. Mathematics, 12.","DOI":"10.3390\/math12040584"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1007\/s10985-022-09576-2","article-title":"Targeted maximum likelihood estimation for causal inference in survival and competing risks analysis","volume":"30","author":"Rytgaard","year":"2024","journal-title":"Lifetime Data Anal."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1007\/s11222-023-10358-5","article-title":"Maximum likelihood estimation for discrete latent variable models via evolutionary algorithms","volume":"34","author":"Luca","year":"2024","journal-title":"Stat. Comput."},{"key":"ref_20","first-page":"505","article-title":"Reliability: Measurement, Theory, and Practice","volume":"36","author":"Goldengorin","year":"2020","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zhuang, L., Xu, A., Wang, Y., and Tang, Y. (2024). Remaining useful life prediction for two-phase degradation model based on reparameterized inverse Gaussian process. Eur. J. Oper. Res.","DOI":"10.1016\/j.ejor.2024.06.032"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Xu, A., Wang, B., Zhu, D., Pang, J., and Lian, X. (2024). Bayesian reliability assessment of permanent magnet brake under small sample size. IEEE Trans. Reliab.","DOI":"10.1109\/TR.2024.3381072"},{"key":"ref_23","first-page":"549","article-title":"The Fr\u00e9chet distribution: Estimation and application\u2014An overview","volume":"23","author":"Pedro","year":"2020","journal-title":"J. Stat. Manag. Syst."},{"key":"ref_24","first-page":"353","article-title":"On Estimation of the Fr\u00b4echet Distribution with Known Shape under Different Sample Schemes","volume":"7","author":"Tabassum","year":"2014","journal-title":"Int. J. Intell. Technol. Appl. Stat."},{"key":"ref_25","unstructured":"Casella, G., and Berger, R.L. (2002). Statistical Inference, Duxbury Press. [2nd ed.]."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data, John Wiley and Sons. [2nd ed.].","DOI":"10.1002\/9781118033005"},{"key":"ref_27","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":"Gail","year":"1978","journal-title":"J. R. Stat. Soc. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1109\/TR.1987.5222310","article-title":"How to identify a bathtub hazard rate","volume":"36","author":"Aarset","year":"1987","journal-title":"IEEE Trans. Reliab."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/8\/1060\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:37:47Z","timestamp":1760110667000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/8\/1060"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,16]]},"references-count":28,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["sym16081060"],"URL":"https:\/\/doi.org\/10.3390\/sym16081060","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,8,16]]}}}