{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T13:56:48Z","timestamp":1784296608884,"version":"3.55.0"},"reference-count":38,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2022,10,9]],"date-time":"2022-10-09T00:00:00Z","timestamp":1665273600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science, Research and Innovation Fund (NSRF), and King Mongkut\u2019s University of Technology North Bangkok","award":["KMUTNB-FF-65-23"],"award-info":[{"award-number":["KMUTNB-FF-65-23"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Birnbaum\u2013Saunders (BS) distribution, also known as the fatigue life distribution, is right-skewed and used to model the failure times of industrial components. It has received much attention due to its attractive properties and its relationship to the normal distribution (which is symmetric). Furthermore, the coefficient of variation (CV) is commonly used to analyze variation within a dataset. In some situations, the independent samples are collected from different instruments or laboratories. Consequently, it is of importance to make inference for the common CV. To this end, confidence intervals based on the generalized confidence interval (GCI), method of variance estimates recovery (MOVER), large-sample (LS), Bayesian credible interval (BayCrI), and highest posterior density interval (HPDI) methods are proposed herein to estimate the common CV of several BS distributions. Their performances in terms of their coverage probabilities and average lengths were investigated by using Monte Carlo simulation. The simulation results indicate that the HPDI-based confidence interval outperformed the others in all of the investigated scenarios. Finally, the efficacies of the proposed confidence intervals are illustrated by applying them to real datasets of PM10 (particulate matter \u2264 10 \u03bcm) concentrations from three pollution monitoring stations in Chiang Mai, Thailand.<\/jats:p>","DOI":"10.3390\/sym14102101","type":"journal-article","created":{"date-parts":[[2022,10,11]],"date-time":"2022-10-11T03:32:56Z","timestamp":1665459176000},"page":"2101","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Confidence Intervals for Common Coefficient of Variation of Several Birnbaum\u2013Saunders Distributions"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2322-5010","authenticated-orcid":false,"given":"Wisunee","family":"Puggard","sequence":"first","affiliation":[{"name":"Department of Applied Statistics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8269-3397","authenticated-orcid":false,"given":"Sa-Aat","family":"Niwitpong","sequence":"additional","affiliation":[{"name":"Department of Applied Statistics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3059-1131","authenticated-orcid":false,"given":"Suparat","family":"Niwitpong","sequence":"additional","affiliation":[{"name":"Department of Applied Statistics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,10,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Vassilopoulos, A.P. (2020). Fatigue life modeling and prediction methods for composite materials and structures\u2014Past, present, and future prospects. Fatigue Life Prediction of Composites and Composite Structures, Woodhead Publishing.","DOI":"10.1016\/B978-0-08-102575-8.00001-2"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"319","DOI":"10.2307\/3212003","article-title":"A new family of life distributions","volume":"6","author":"Birnbaum","year":"1969","journal-title":"J. Appl. Probab."},{"key":"ref_3","first-page":"103","article-title":"Birnbaum\u2013Saunders and lognormal kernel estimators for modelling durations in high frequency financial data","volume":"4","author":"Jin","year":"2003","journal-title":"Ann. Econ. Financ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1002\/bimj.201600008","article-title":"Birnbaum\u2013Saunders frailty regression models: Diagnostics and application to medical data","volume":"59","author":"Leao","year":"2017","journal-title":"Biom. J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1080\/03610910903350508","article-title":"Acceptance sampling plans from truncated life tests based on the Birnbaum\u2013Saunders distribution for percentiles","volume":"39","author":"Lio","year":"2010","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1479","DOI":"10.1007\/s00477-012-0684-8","article-title":"A nonparametric method for estimating asymmetric densities based on skewed Birnbaum\u2013Saunders distributions applied to environmental data","volume":"27","author":"Saulo","year":"2013","journal-title":"Stoch. Environ. Res. Risk. Assess."},{"key":"ref_7","unstructured":"Rice, J.A. (2006). Mathematical Statistics and Data Analysis, Duxbury Press."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2213","DOI":"10.1002\/sim.2088","article-title":"Inferences on the common coefficient of variation","volume":"24","author":"Tian","year":"2005","journal-title":"Stat. Med."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2187","DOI":"10.1080\/03610920701215126","article-title":"Confidence bounds and hypothesis tests for normal distribution coefficients of variation","volume":"36","author":"Verrill","year":"2007","journal-title":"Commun. Statist. Theor. Meth."},{"key":"ref_10","first-page":"349","article-title":"Generalized confidence interval for the common coefficient of variation","volume":"7","author":"Behboodian","year":"2008","journal-title":"J. Stat. Theory Appl."},{"key":"ref_11","first-page":"117","article-title":"Inference on the common coefficient of variation when populations are lognormal: A simulation-based approach","volume":"11","author":"Ng","year":"2014","journal-title":"J. Stat. Adv. Theory Appl."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Thangjai, W., and Niwitpong, S.A. (2017). Confidence intervals for the weighted coefficients of variation of two-parameter exponential distributions. Cogent Math., 4.","DOI":"10.1080\/23311835.2017.1315880"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1214\/15-EJS993","article-title":"A note on combined inference on the common coefficient of variation using confidence distributions","volume":"9","author":"Liu","year":"2015","journal-title":"Electron. J. Stat."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Yosboonruang, N., Niwitpong, S.A., and Niwitpong, S. (2022). Bayesian computation for the common coefficient of variation of delta-lognormal distributions with application to common rainfall dispersion in Thailand. PeerJ, 10.","DOI":"10.7717\/peerj.12858"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"328","DOI":"10.2307\/3212004","article-title":"Estimation for a family of life distributions with applications to fatigue","volume":"6","author":"Birnbaum","year":"1969","journal-title":"J. Appl. Probab."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"251","DOI":"10.2307\/1267788","article-title":"Inference on the parameters of the Birnbaum\u2013Saunders fatigue life distribution based on maximum likelihood estimation","volume":"23","author":"Engelhardt","year":"1981","journal-title":"Technometrics"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1016\/S0167-9473(02)00254-2","article-title":"Modified moment estimation for the two-parameter Birnbaum\u2013Saunders distribution","volume":"43","author":"Ng","year":"2003","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1016\/j.csda.2003.11.018","article-title":"Improved interval estimation for the two-parameter Birnbaum-Saunders distribution","volume":"47","author":"Wu","year":"2004","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.csda.2009.08.004","article-title":"Reference analysis for Birnbaum-Saunders distribution","volume":"54","author":"Xu","year":"2010","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4320","DOI":"10.1016\/j.csda.2012.03.023","article-title":"Generalized interval estimation for the Birnbaum-Saunders distribution","volume":"56","author":"Wang","year":"2012","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s00180-015-0629-z","article-title":"Bayesian analysis of Birnbaum-Saunders distribution via the generalized ratio-of-uniforms method","volume":"31","author":"Wang","year":"2016","journal-title":"Comput. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1080\/02664763.2016.1189521","article-title":"Inference for the common mean of several Birnbaum\u2013Saunders populations","volume":"44","author":"Guo","year":"2017","journal-title":"J. Appl. Stat."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Puggard, W., Niwitpong, S.A., and Niwitpong, S. (2021). Bayesian estimation for the coefficients of variation of Birnbaum\u2013Saunders distributions. Symmetry, 13.","DOI":"10.3390\/sym13112130"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Puggard, W., Niwitpong, S.A., and Niwitpong, S. (2022). Confidence intervals for comparing the variances of two independent Birnbaum\u2013Saunders distributions. Symmetry, 14.","DOI":"10.3390\/sym14071492"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"899","DOI":"10.1080\/01621459.1993.10476355","article-title":"Generalized confidence intervals","volume":"88","author":"Weerahandi","year":"1993","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_26","unstructured":"Weerahandi, S. (2004). Generalized Inference in Repeated Measures: Exact Methods in MANOVA and Mixed Models, Wiley."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1080\/00401706.2015.1096827","article-title":"Estimation of field reliability based on aggregate lifetime data","volume":"59","author":"Chen","year":"2017","journal-title":"Technometrics"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1080\/00224065.2017.11918185","article-title":"Approximate statistical limits for a gamma distribution","volume":"49","author":"Chen","year":"2017","journal-title":"J. Qaul. Tech."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1080\/00224065.2018.1436839","article-title":"Uncertainty quantification for monotone stochastic degradation models","volume":"50","author":"Chen","year":"2018","journal-title":"J. Qaul. Tech."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Luo, C., Shen, L., and Xu, A. (2022). Modelling and estimation of system reliability under dynamic operating environments and lifetime ordering constraints. Rel. Eng. Sys. Safe., 218.","DOI":"10.1016\/j.ress.2021.108136"},{"key":"ref_31","first-page":"1558","article-title":"The confidence intervals for the scale parameter of the Birnbaum\u2013Saunders fatigue life distribution","volume":"30","author":"Sun","year":"2009","journal-title":"Acta Armamentarii"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1693","DOI":"10.1002\/sim.3095","article-title":"Construction of confidence limits about effect measures: A general approach","volume":"27","author":"Zou","year":"2008","journal-title":"Stat. Med."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3755","DOI":"10.1016\/j.csda.2009.03.016","article-title":"Confidence interval estimation for lognormal data with application to health economics","volume":"53","author":"Zou","year":"2009","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2919","DOI":"10.1177\/0962280215615160","article-title":"Standardized likelihood ratio test for comparing several log-normal means and confidence interval for the common mean","volume":"26","author":"Krishnamoorthy","year":"2015","journal-title":"Stat. Methods Med. Res."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2324","DOI":"10.1016\/j.csda.2011.01.021","article-title":"Bayesian analysis of Birnbaum\u2013Saunders distribution with partial information","volume":"55","author":"Xu","year":"2011","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/BF01889987","article-title":"Efficient generation of random variates via the ratio-of-uniforms method","volume":"1","author":"Wakefield","year":"1991","journal-title":"Stat. Comput."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Box, G.E.P., and Tiao, G.C. (1992). Bayesian Inference in Statistical Analysis, Wiley.","DOI":"10.1002\/9781118033197"},{"key":"ref_38","unstructured":"(2022, July 25). Reports on Smog Situation in the North Home Page, (In Thai)."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2101\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:48:50Z","timestamp":1760143730000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/10\/2101"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,9]]},"references-count":38,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2022,10]]}},"alternative-id":["sym14102101"],"URL":"https:\/\/doi.org\/10.3390\/sym14102101","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,9]]}}}