{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T07:17:13Z","timestamp":1782976633990,"version":"3.54.5"},"reference-count":26,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,28]],"date-time":"2025-06-28T00:00:00Z","timestamp":1751068800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science, Research, and Innovation Fund (NSRF)","award":["KMUTNB-FF-68-B-43"],"award-info":[{"award-number":["KMUTNB-FF-68-B-43"]}]},{"name":"King Mongkut\u2019s University of Technology North Bangkok","award":["KMUTNB-FF-68-B-43"],"award-info":[{"award-number":["KMUTNB-FF-68-B-43"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Rayleigh distribution is a continuous probability distribution that is inherently asymmetric and commonly used to model right-skewed data. It holds significant importance across a wide range of scientific and engineering disciplines and exhibits structural relationships with several other asymmetric probability distributions, for example, Weibull and exponential distribution. This research proposes techniques for establishing credible intervals and confidence intervals for the single mean of the zero-inflated two-parameter Rayleigh distribution. The study introduces methods such as the percentile bootstrap, generalized confidence interval, standard confidence interval, approximate normal using the delta method, Bayesian credible interval, and Bayesian highest posterior density. The effectiveness of the proposed methods is assessed by evaluating coverage probability and expected length through Monte Carlo simulations. The results indicate that the Bayesian highest posterior density method outperforms the other approaches. Finally, the study applies the proposed methods to construct confidence intervals for the single mean using real-world data on COVID-19 total deaths in Singapore during October 2022.<\/jats:p>","DOI":"10.3390\/sym17071019","type":"journal-article","created":{"date-parts":[[2025,6,30]],"date-time":"2025-06-30T03:54:28Z","timestamp":1751255668000},"page":"1019","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Confidence Intervals for the Parameter Mean of Zero-Inflated Two-Parameter Rayleigh Distribution"],"prefix":"10.3390","volume":"17","author":[{"given":"Sasipong","family":"Kijsason","sequence":"first","affiliation":[{"name":"Department of Applied Statistics, Faculty of Applied Sciences, 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 Sciences, 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 Sciences, King MongKut\u2019s University of Technology North Bangkok, Bangkok 10800, Thailand"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1080\/14786449708620990","article-title":"On the resultant of a large number of vibrations of the same pitch and of arbitrary phase","volume":"43","author":"Rayleigh","year":"1880","journal-title":"Philos. Mag. J. Sci."},{"key":"ref_2","first-page":"1907","article-title":"The analysis of wind data with Rayleigh distribution and optimum turbine and cost analysis in Elmadag, Turkey","volume":"15","author":"Arikan","year":"2015","journal-title":"Istanb.-Univ.-J. Electr. Electron. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Almongy, H.M., Almetwally, E.M., Aljohani, H.M., Alghamdi, A.S., and Hafez, E.H. (2021). A new extended Rayleigh distribution with applications of COVID-19 data. J. Results Phys., 23.","DOI":"10.1016\/j.rinp.2021.104012"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Kilai, M., Waititu, G.A., Kibira, W.A., Abd El-Raouf, M.M., and Abushal, T.A. (2022). A new versatile modification of the Rayleigh distribution for modeling COVID-19 mortality rates. J. Results Phys., 35.","DOI":"10.1016\/j.rinp.2022.105260"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.18576\/jsap\/120316","article-title":"Power Rayleigh Distribution for Fitting Total Deaths of COVID-19 in Egypt","volume":"12","author":"Kilany","year":"2023","journal-title":"J. Stat. Appl. Prob."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"219","DOI":"10.18187\/pjsor.v19i2.4130","article-title":"Bayesian Estimation in Rayleigh Distribution under a Distance Type Loss Function","volume":"19","author":"Seal","year":"2023","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1080\/24725854.2024.2389538","article-title":"A multivariate student-t process model for dependent tail-weighted degradation data","volume":"57","author":"Xu","year":"2024","journal-title":"IISE Trans."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Xu, A., Wang, R., Weng, X., Wu, Q., and Zhuang, L. (2025). Strategic integration of adaptive sampling and ensemble techniques in federated learning for aircraft engine remaining useful life prediction. Appl. Soft Comput., 175.","DOI":"10.1016\/j.asoc.2025.113067"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1080\/24754269.2025.2495651","article-title":"A Bayesian hierarchical model with spatially varying dispersion for reference-free cell type deconvolution in spatial transcriptomics","volume":"9","author":"Li","year":"2025","journal-title":"Stat. Theory Relat. Fields"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1080\/24754269.2025.2483553","article-title":"An algorithm for distributed parameter estimation in modal regression models","volume":"9","author":"Ma","year":"2025","journal-title":"Stat. Theory Relat. Fields"},{"key":"ref_11","first-page":"55","article-title":"Two-parameter Rayleigh distribution: Different Methods of Estimation","volume":"33","author":"Dey","year":"2014","journal-title":"Am. J. Math. Manag. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ardianti, F. (2018). Estimating parameter of Rayleigh distribution by using maximum likelihood method and Bayes method. Iop Conf. Ser. Mater. Sci. Eng., 300.","DOI":"10.1088\/1757-899X\/300\/1\/012036"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1067","DOI":"10.18576\/amis\/140614","article-title":"Estimating the parameters of reliability function based on Rayleigh distribution","volume":"14","author":"Mustafa","year":"2020","journal-title":"Appl. Math Inf. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1080\/02664763.2019.1634681","article-title":"Confidence interval, prediction interval and tolerance limits for a two-parameter Rayleigh distribution","volume":"47","author":"Krishnamoorthy","year":"2020","journal-title":"J. Appl. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/1269547","article-title":"Zero-inflated Poisson regression with an application to defects in manufacturing","volume":"34","author":"Lambert","year":"1992","journal-title":"Technometrics"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Liu, F., Liu, J., and Xu, P. (2023). A novel zero-inflated Rayleigh distribution and its properties. Results Phys., 51.","DOI":"10.1016\/j.rinp.2023.106634"},{"key":"ref_17","first-page":"439","article-title":"Statistical Estimation of mean of delta-lognormal distribution","volume":"18","author":"Maneerat","year":"2020","journal-title":"Thail. Stat."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Seo, J., Jeon, J., and Kang, S. (2016). Exact interval inference for the two-parameter Rayleigh distribution based on the upper record values. J. Probab. Stat., 2016.","DOI":"10.1155\/2016\/8246390"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Teawpongpan, T., and Sangnawakij, P. (2023). Confidence interval for the population mean of Rayleigh distribution with application to shear wave velocity of soils in Thailand. J. Appl. Sci. Emerg. Technol., 22.","DOI":"10.14416\/JASET.KMUTNB.2023.03.006"},{"key":"ref_20","first-page":"3740","article-title":"Analysis of a new jointly hybrid censored Rayleigh populations","volume":"9","author":"Elshahhat","year":"2024","journal-title":"JAIMS Math."},{"key":"ref_21","first-page":"202","article-title":"The rayleigh lindley distribution: A new generalization of rayleigh distribution with physical applications","volume":"44","author":"Ahmad","year":"2023","journal-title":"Rev. Investig. Oper."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"11523","DOI":"10.1016\/j.aej.2022.05.010","article-title":"A new generalized Rayleigh distribution with analysis to big data of an online community","volume":"61","author":"Shen","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Efron, B., and Tibshirani, R.J. (1993). An Introduction to the Bootstrap, Chapman and Hall CRC.","DOI":"10.1007\/978-1-4899-4541-9"},{"key":"ref_24","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."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1471","DOI":"10.1080\/02664763.2014.881780","article-title":"Generalized confidence interval estimation for the mean of delta-lognormal distribution: An application to New Zealand trawl survey data","volume":"41","author":"Wu","year":"2014","journal-title":"J. Appl. Stat."},{"key":"ref_26","unstructured":"Box, G., and Tiao, G.C. (1973). Bayesian Inference in Statistical Analysis, Wiley Classics."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/7\/1019\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:00:47Z","timestamp":1760032847000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/7\/1019"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,28]]},"references-count":26,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["sym17071019"],"URL":"https:\/\/doi.org\/10.3390\/sym17071019","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,28]]}}}