{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:00:23Z","timestamp":1760148023790,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,21]],"date-time":"2023-03-21T00:00:00Z","timestamp":1679356800000},"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 111-2118-M-032-003-MY2"],"award-info":[{"award-number":["NSTC 111-2118-M-032-003-MY2"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Making use of uniformly minimum-variance unbiased estimators for the parameters of two-parameter exponential distributions and the distribution of pivotal quantities, we propose one-stage multiple comparison procedures for k mean lifetimes with the average under heteroscedasticity. The multiple comparison procedures include one-sided and two-sided confidence intervals. These intervals can be applied to identify which treatment\u2019s mean lifetime is better than the average or worse than the average in terms of the mean lifetimes of all treatments. Critical values are obtained in order to assure users that the given confidence coefficient has been reached; they are organized in table format for practical and convenient use. An example is provided to demonstrate the proposed techniques, wherein the mean survival times of four different lung cancer categories are compared with the average.<\/jats:p>","DOI":"10.3390\/axioms12030312","type":"journal-article","created":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T06:35:28Z","timestamp":1679466928000},"page":"312","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["One-Stage Multiple Comparisons of the Mean Lifetimes of k Treatments with the Average for Exponential Distributions under Heteroscedasticity"],"prefix":"10.3390","volume":"12","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"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1214\/aoms\/1177728845","article-title":"A single sample multiple decision procedure for ranking means of normal populations with known variances","volume":"25","author":"Bechhofer","year":"1954","journal-title":"Ann. Math. Stat."},{"key":"ref_2","unstructured":"Gupta, S.S. (1956). On a Decision Rule for a Problem in Ranking Means. [Ph.D. Thesis, Institute of Statistics, University of North Carolina]. Mimeograph Series No. 150."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data, Wiley.","DOI":"10.1002\/9781118033005"},{"key":"ref_4","unstructured":"Johnson, N.L., Kotz, S., and Balakrishnan, N. (1994). Continuous Univariate Distributions, Wiley."},{"key":"ref_5","unstructured":"Bain, L.J., and Engelhardt, M. (1991). Statistical Analysis of Reliability and Life Testing Models, Marcel Dekker."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1002\/nav.3800270215","article-title":"Analysis of data from life test experiments under an exponential model","volume":"27","author":"Lawless","year":"1980","journal-title":"Nav. Res. Logist. Q."},{"key":"ref_7","unstructured":"Balakrishnan, K. (1996). Exponential Distribution Theory, Methods and Applications, Routledge."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1002\/nav.3800310105","article-title":"Product moments of order statistics from the doubly truncated exponential distribution","volume":"31","author":"Balakrishnan","year":"1984","journal-title":"Nav. Res. Logist. Q."},{"key":"ref_9","first-page":"1","article-title":"Best linear unbiased and maximum likelihood estimation for exponential distributions under general progressive type-II censored samples","volume":"58","author":"Balakrishnan","year":"1996","journal-title":"Sankhy\u0101 Indian J. Stat. Ser. B."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"303","DOI":"10.2991\/jsta.d.190818.004","article-title":"Characterization of exponential distribution through normalized spacing of generalized order statistics","volume":"18","author":"Khan","year":"2019","journal-title":"J. Stat. Theory Appl."},{"key":"ref_11","first-page":"383","article-title":"Multiple comparison of exponential location parameters with the best under type II censoring","volume":"12","author":"Ng","year":"1993","journal-title":"Am. J. Math. Manag. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1080\/07474949008836202","article-title":"Two-stage procedures for comparing several exponential populations with a control when the scale parameters are unknown and unequal","volume":"9","author":"Lam","year":"1990","journal-title":"Seq. Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1372","DOI":"10.1016\/j.csda.2009.12.002","article-title":"One-stage multiple comparisons with the control for exponential location parameters under heteroscedasticity","volume":"54","author":"Wu","year":"2010","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1080\/03610918.2010.549988","article-title":"Multiple comparisons with more than one control for exponential location parameters under heteroscedasticity","volume":"40","author":"Maurya","year":"2011","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1817","DOI":"10.1080\/02664763.2013.796350","article-title":"Multiple comparisons with a control for exponential location parameters under heteroscedasticity","volume":"40","author":"Maurya","year":"2013","journal-title":"J. Appl. Stat."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2968","DOI":"10.1080\/03610918.2017.1366515","article-title":"One stage multiple comparisons of k-1 treatment mean lifetimes with the control for exponential distributions under heteroscedasticity","volume":"47","author":"Wu","year":"2018","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1473","DOI":"10.1080\/03610918.2019.1584302","article-title":"One stage multiple comparisons with the control for exponential mean lifetimes based on doubly censored samples under heteroscedasticity","volume":"50","author":"Wu","year":"2021","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Wu, S.F. (2022). Multiple comparison procedures for exponential mean lifetimes compared with several controls. Mathematics, 10.","DOI":"10.3390\/math10040609"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2740","DOI":"10.1080\/00949655.2015.1124433","article-title":"New one-stage multiple comparisons procedures with the average for exponential location parameters under heteroscedasticity","volume":"86","author":"Wu","year":"2016","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_20","unstructured":"Lam, K. (, January August). Subset selection of normal populations under heteroscedasticity. Proceedings of the IPASRAS-II Proceedings and Discussions of the Second International Conference on Inference Procedures Associated with Statistal Ranking and Selection on The frontiers of Modern Statistical Inference Procedures, II, Sydney, Australia."},{"key":"ref_21","unstructured":"Roussas, G.G. (2014). A Course in Mathematical Statistics, Elsevier Science & Technology Books. [3rd ed.]."},{"key":"ref_22","unstructured":"Mourik, T.V. (2005). Fortran 90\/95 Programming Manual, Chemistry Department, University College London."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/3\/312\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:59:54Z","timestamp":1760122794000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/3\/312"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,21]]},"references-count":22,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["axioms12030312"],"URL":"https:\/\/doi.org\/10.3390\/axioms12030312","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,3,21]]}}}