{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T12:49:38Z","timestamp":1770986978637,"version":"3.50.1"},"reference-count":46,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,24]],"date-time":"2022-12-24T00:00:00Z","timestamp":1671840000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Mongkut\u2019s University of Technology North Bangkok","award":["KMUTNB-PHD-62-04"],"award-info":[{"award-number":["KMUTNB-PHD-62-04"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Weibull distribution, one of the most significant distributions with applications in numerous fields, is associated with numerous distributions such as generalized gamma distribution, exponential distribution, and Rayleigh distribution, which are asymmetric. Nevertheless, it shares a close relationship with a normal distribution where a process of transformation allows them to become symmetric. The Weibull distribution is commonly used to study the failure of components and phenomena. It has been applied to a variety of scenarios, including failure time, claims amount, unemployment duration, survival time, and especially wind speed data. A suitable area for installing a wind turbine requires a wind speed that is both sufficiently high and consistent, and so comparing the variation in wind speed in two areas is eminently desirable. In this paper, methods to estimate the confidence interval for the ratio of the coefficients of variation of two Weibull distributions are proposed and applied to compare the variation in wind speed in two areas. The methods are the generalized confidence interval (GCI), the method of variance estimates recovery (MOVER), and Bayesian methods based on the gamma and uniform priors. The Bayesian methods comprise the equal-tailed confidence interval and the highest posterior density (HPD) interval. The effectiveness of the methods was evaluated in terms of their coverage probabilities and expected lengths and also empirically applied to wind speed datasets from two different areas in Thailand. The results indicate that the HPD interval based on the uniform prior outperformed the others in most of the scenarios tested and so it is suggested for estimating the confidence interval for the ratio of the coefficients of variation of two Weibull distributions.<\/jats:p>","DOI":"10.3390\/sym15010046","type":"journal-article","created":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T04:58:01Z","timestamp":1672117081000},"page":"46","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Estimation of the Confidence Interval for the Ratio of the Coefficients of Variation of Two Weibull Distributions and Its Application to Wind Speed Data"],"prefix":"10.3390","volume":"15","author":[{"given":"Manussaya","family":"La-ongkaew","sequence":"first","affiliation":[{"name":"Department of Applied Statistics, Faculty of Applied Science, King Mongkut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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":[{"role":"author","vocabulary":"crossref"}]},{"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":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"863274","DOI":"10.1155\/2011\/863274","article-title":"A simple normal approximation for Weibull distribution with application to estimation of upper prediction limit","volume":"2011","author":"Kulkarni","year":"2011","journal-title":"J. Probab. Stat."},{"key":"ref_2","first-page":"299","article-title":"Degree of tuberculin sensitivity in guinea pigs infected with different doses of virulent tubercle bacilli","volume":"71","author":"Bjerkdal","year":"1960","journal-title":"Am. J. Hyg."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"108136","DOI":"10.1016\/j.ress.2021.108136","article-title":"Modelling and estimation of system reliability under dynamic operating environments and lifetime ordering constraints","volume":"218","author":"Luo","year":"2022","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_4","first-page":"65","article-title":"A unified model for system reliability evaluation under dynamic operating conditions","volume":"7","author":"Xu","year":"2019","journal-title":"IEEE Trans. Reliab."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1198\/TECH.2010.08210","article-title":"Inference under progressively type II right-censored sampling for certain lifetime distributions","volume":"52","author":"Wang","year":"2010","journal-title":"Technometrics"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Zhang, L., Xu, A., An, L., and Li, M. (2022). Bayesian inference of system reliability for multicomponent stress-strength model under Marshall-Olkin Weibull distribution. Systems, 10.","DOI":"10.3390\/systems10060196"},{"key":"ref_7","first-page":"1","article-title":"Data analysis of progressive-stress accelerated life tests with group effects","volume":"19","author":"Zhuang","year":"2022","journal-title":"Qual. Technol. Quant. Manag."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1476","DOI":"10.1175\/1520-0450(2001)040<1476:EOWSDU>2.0.CO;2","article-title":"Estimation of wind speed distribution using Markov chain Monte Carlo techniques","volume":"2001 40","author":"Pang","year":"2001","journal-title":"J. Appl. Meteorol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1260\/014459806779367509","article-title":"Wind potential assessment using the Weibull model at the Inner Mongolia of China","volume":"24","author":"Jiang","year":"2006","journal-title":"Energy Explor. Exploit."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1109\/24.406588","article-title":"Modeling failure-data by mixture of 2 Weibull distributions: A graphical approach","volume":"44","author":"Jiang","year":"1995","journal-title":"IEEE Trans. Reliab."},{"key":"ref_11","first-page":"708","article-title":"A new approach for Weibull modeling for reliability life data analysis","volume":"250","author":"Elmahdy","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1016\/j.ress.2018.08.008","article-title":"Bayesian estimation of Weibull mixture in heavily censored data setting","volume":"180","author":"Ducros","year":"2018","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_13","first-page":"10","article-title":"An analysis of wind potential at huasai in nakhon si thammarat province","volume":"12","author":"Chancham","year":"2009","journal-title":"ASEAN J. Sci. Technol. Rep."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1475","DOI":"10.1007\/s12206-010-0415-3","article-title":"Wind power potential and characteristic analysis of Chiang Mai, Thailand","volume":"24","author":"Chaichana","year":"2010","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_15","first-page":"51","article-title":"An analysis of wind speed distribution at Thasala, Nakhon Si Thammarat, Thailand","volume":"2","author":"Waewsak","year":"2011","journal-title":"J. Sustain. Energy Environ."},{"key":"ref_16","unstructured":"Chaichana, T., Chaiyatep, S., Jompakdee, W., Klongpanit, W., and Dussadee, N. (2008, January 28\u201329). Analysis of wind power potential in the rainy season at the high agricultural project under the royal initiative at mon lan using weibull distribution. Proceedings of the 2nd Technology and Innovation for Sustainable Development Conference, Khon Kaen, Thailand."},{"key":"ref_17","first-page":"41","article-title":"Evaluation of Wind Energy Potential and Electricity Generation in Northern of Thailand","volume":"24","author":"Sasujit","year":"2016","journal-title":"Naresuan Univ. Journal: Sci. Technol. (NUJST)"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"380","DOI":"10.1016\/j.egypro.2017.10.176","article-title":"Wind turbine and local acceptance in Southern Thailand","volume":"138","author":"Kongprasit","year":"2017","journal-title":"Energy Procedia"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1377\/hlthaff.12.1.162","article-title":"Impact of socioeconomic status on hospital use in New York City","volume":"12","author":"Billings","year":"1993","journal-title":"Health Aff."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.ijhydene.2004.03.041","article-title":"An investigation on the causes of cycle variation in direct injection hydrogen fueled engines","volume":"30","author":"Kim","year":"2005","journal-title":"Int. J. Hydrogen Energy"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1590\/S1678-77572005000300008","article-title":"Analysis of the coefficient of variation in shear and tensile bond strength tests","volume":"13","author":"Romano","year":"2005","journal-title":"J. Appl. Oral Sci."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"440","DOI":"10.2306\/scienceasia1513-1874.2013.39.440","article-title":"An examination of the variability of agricultural production using a new approximate confidence interval for the coefficient of variation","volume":"39","author":"Saelee","year":"2013","journal-title":"ScienceAsia"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"43","DOI":"10.3389\/fams.2019.00043","article-title":"Performance of some estimators of relative variability","volume":"5","author":"Ospina","year":"2005","journal-title":"Front. Appl. Math. Stat."},{"key":"ref_24","unstructured":"Casella, G., and Berger, R.L. (2002). Statistical Inference, Duxbury Press. [2nd ed.]."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"e9662","DOI":"10.7717\/peerj.9662","article-title":"The Bayesian confidence intervals for measuring the difference between dispersion of rainfall in Thailand","volume":"8","author":"Yosboonruang","year":"2020","journal-title":"PeerJ"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"e11676","DOI":"10.7717\/peerj.11676","article-title":"Confidence intervals for the difference between the coefficients of variation of Weibull distributions for analyzing wind speed dispersion","volume":"9","author":"Niwitpong","year":"2021","journal-title":"PeerJ"},{"key":"ref_27","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.-Stat.\u2013Theory Methods"},{"key":"ref_28","first-page":"3811","article-title":"Confidence intervals for the ratio of coefficients of variation of delta-lognormal distribution","volume":"7","author":"Buntao","year":"2013","journal-title":"Appl. Math. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Sangnawakij, P., Niwitpong, S.A., and Niwitpong, S. (2015, January 15\u201317). Confidence intervals for the ratio of coefficients of variation of the gamma distributions. Proceedings of the International Symposium on Integrated Uncertainty in Knowledge Modelling and Decision Making, Nha Trang, Vietnam.","DOI":"10.1007\/978-3-319-25135-6_19"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"13","DOI":"10.14445\/22315373\/IJMTT-V29P503","article-title":"Confidence intervals for the difference and the ratio of coefficients of variation of normal distribution with a known ratio of variances","volume":"29","author":"Niwitpong","year":"2016","journal-title":"Int. J. Math. Trends Technol."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Sangnawakij, P., Niwitpong, S.A., and Niwitpong, S. (2016, January 30). Confidence intervals for the ratio of coefficients of variation in the two-parameter exponential distributions. Proceedings of the International Symposium on Integrated Uncertainty in Knowledge Modelling and Decision Making, Da Nang, Vietnam.","DOI":"10.1007\/978-3-319-49046-5_46"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"345","DOI":"10.2991\/jsta.2017.16.3.6","article-title":"Improved Confidence Intervals for the Ratio of Coefficients of Variation of Two Lognormal Distributions","volume":"16","author":"Hasan","year":"2017","journal-title":"J. Stat. Theory Appl."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Puggard, W., Niwitpong, S.A., and Niwitpong, S. (2020, January 11\u201313). Generalized Confidence Interval of the Ratio of Coefficients of Variation of Birnbaum-Saunders Distribution. Proceedings of the International Symposium on Integrated Uncertainty in Knowledge Modelling and Decision Making, Phuket, Thailand.","DOI":"10.1007\/978-3-030-62509-2_33"},{"key":"ref_34","first-page":"489","article-title":"Statistical inference on the ratio of delta-lognormal coefficients of variation","volume":"14","author":"Yosboonruang","year":"2021","journal-title":"Appl. Sci. Eng. Prog."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1080\/00401706.1965.10490300","article-title":"Maximum likelihood estimation in the Weibull distribution based on complete and on censored samples","volume":"7","author":"Cohen","year":"1965","journal-title":"Technometrics"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"247","DOI":"10.2307\/1268358","article-title":"Maximum likelihood estimation for the three parameter Weibull distribution based on censored samples","volume":"17","author":"Lemon","year":"1975","journal-title":"Technometrics"},{"key":"ref_37","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_38","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1007\/s00184-006-0074-7","article-title":"Inference on reliability in two-parameter exponential stress\u2013strength model","volume":"65","author":"Krishnamoorthy","year":"2007","journal-title":"Metrika"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1177\/0962280210383082","article-title":"Closed-form confidence intervals for functions of the normal mean and standard deviation","volume":"21","author":"Donner","year":"2010","journal-title":"Stat. Methods Med. Res."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1214\/aoms\/1177732503","article-title":"The sampling distribution of the coefficient of variation","volume":"7","author":"Hendricks","year":"2010","journal-title":"Ann. Math. Stat."},{"key":"ref_41","first-page":"721","article-title":"Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images","volume":"6","author":"Geman","year":"2010","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Kruschke, J.K. (2015). Chapter 12-Bayesian Approaches to Testing a Point (\u201cNull\u201d) Hypothesis in 478 Doing Bayesian Data Analysis, Academic Press.","DOI":"10.1016\/B978-0-12-405888-0.00012-X"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Khan, M., and Ahmed, M.R. (2022). Bayesian method for estimating Weibull parameters for wind resource assessment in the Equatorial region: A comparison between two-parameter and three-parameter Weibull distributions. Wind Energy Sci. Discuss., 1\u201338.","DOI":"10.21203\/rs.3.rs-504670\/v4"},{"key":"ref_44","unstructured":"Using R to Run \u2018JAGS\u2019 (2022, June 15). R package Version 0.6-1. Available online: https:\/\/cran.rproject.org\/web\/packages\/R2jags."},{"key":"ref_45","unstructured":"(2022, January 10). Department of Alternative Energy Development and Efficiency, Ministry of Energy, Available online: https:\/\/www.dede.go.th."},{"key":"ref_46","unstructured":"(2022, February 07). Report of Wind Resource Assessment of Thailand, Available online: http:\/\/www2.dede.go.th\/dede\/renew\/Twm\/REPORT.htm."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/46\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:50:03Z","timestamp":1760147403000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/46"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,24]]},"references-count":46,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["sym15010046"],"URL":"https:\/\/doi.org\/10.3390\/sym15010046","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,24]]}}}