{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T09:33:37Z","timestamp":1763026417862,"version":"3.45.0"},"reference-count":25,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T00:00:00Z","timestamp":1762992000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science, Research, and Innovation Fund","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>This study addresses the estimation of the common mean for the zero-inflated inverse Gaussian (ZIIG) distributions, a problem not previously explored. The performance of four interval estimation approaches was evaluated: the generalized confidence interval (GCI), parametric bootstrap, Bayesian, and highest posterior density (HPD). Simulation studies under varying sample sizes, zero-inflation probabilities, mean values, and shape parameters revealed notable differences in coverage probability (CP) and average length (AL). For small samples, the GCI and parametric bootstrap approaches often under-covered, particularly in highly skewed or heavily zero-inflated cases. In contrast, Bayesian and HPD intervals generally maintained coverage closer to the nominal 0.95 level, albeit with longer intervals. As sample size increased, all methods approached nominal coverage and produced shorter intervals, improving precision. Overall, the Bayesian and HPD approaches demonstrated strong robustness across conditions, with HPD intervals frequently achieving accurate coverage with shorter lengths. Finally, the proposed approaches were applied to real-world data on road accident fatalities in Thailand.<\/jats:p>","DOI":"10.3390\/sym17111944","type":"journal-article","created":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T09:10:45Z","timestamp":1763025045000},"page":"1944","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Estimation of the Common Mean of Zero-Inflated Inverse Gaussian Distributions: Application to Road Accident Fatalities in Thailand"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9306-3742","authenticated-orcid":false,"given":"Warisa","family":"Thangjai","sequence":"first","affiliation":[{"name":"Department of Statistics, Faculty of Science, Ramkhamhaeng University, Bangkok 10240, 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":[[2025,11,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2082","DOI":"10.1016\/j.jspi.2007.09.005","article-title":"Inferences on the difference and ratio of the means of two inverse Gaussian distributions","volume":"138","author":"Krishnamoorthy","year":"2008","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1260","DOI":"10.1080\/02664763.2018.1542668","article-title":"A new look at the inverse Gaussian distribution with applications to insurance and economic data","volume":"46","author":"Punzo","year":"2019","journal-title":"J. Appl. Stat."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Gelman, A., and Hill, J. (2007). Data Analysis Using Regression and Multilevel\/Hierarchical Models, Cambridge University Press.","DOI":"10.32614\/CRAN.package.arm"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Borenstein, M., Hedges, L.V., Higgins, J.P.T., and Rothstein, H.R. (2009). Introduction to Meta-Analysis, John Wiley & Sons, Ltd.","DOI":"10.1002\/9780470743386"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.jmp.2016.03.007","article-title":"How hierarchical models improve point estimates of individual-level parameters","volume":"73","author":"Katahira","year":"2016","journal-title":"J. Math. Psychol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1080\/24754269.2018.1530903","article-title":"Combining estimators of a common parameter across samples","volume":"2","author":"Slud","year":"2018","journal-title":"Stat. Theory Relat. Fields"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"906","DOI":"10.1016\/j.csda.2009.09.039","article-title":"Inferences on the common mean of several inverse Gaussian populations","volume":"54","author":"Ye","year":"2010","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/s00184-021-00829-y","article-title":"On estimating common mean of several inverse Gaussian distributions","volume":"85","author":"Bera","year":"2022","journal-title":"Metrika"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3302","DOI":"10.1080\/03610918.2022.2102652","article-title":"Confidence intervals of difference and ratio of means for zero-adjusted inverse Gaussian distributions","volume":"53","author":"Jana","year":"2022","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Khumpasee, W., Niwitpong, S.A., and Niwitpong, S. (2024). Confidence intervals for the coefficient of variation in delta-inverse Gaussian distributions. Symmetry, 16.","DOI":"10.3390\/sym16111488"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"012009","DOI":"10.1088\/1742-6596\/3042\/1\/012009","article-title":"Confidence intervals for the variance of delta-inverse Gaussian distribution with application to traffic mortality count","volume":"3041","author":"Niwitpong","year":"2025","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1111\/1541-0420.00030","article-title":"Inference on the common means of several normal populations based on the generalized variable method","volume":"59","author":"Krishnamoorthy","year":"2003","journal-title":"Biometrics"},{"key":"ref_14","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_15","doi-asserted-by":"crossref","unstructured":"Chen, Y.H., and Zhou, X.H. (2006). Generalized Confidence Intervals for the Ratio or Difference of Two Means for Lognormal Populations with Zeros, Elsevier.","DOI":"10.1002\/sim.2504"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"944","DOI":"10.1002\/bimj.200710391","article-title":"Inferences on the common mean of several log-normal populations: The generalized variable approach","volume":"49","author":"Tian","year":"2007","journal-title":"Biometrical J."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1080\/0094965021000040884","article-title":"Lower confidence bounds for percentiles of Weibull and Birnbaum-Saunders distributions","volume":"73","author":"Padgett","year":"2003","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2138","DOI":"10.1080\/03610918.2018.1435800","article-title":"Bootstrap confidence intervals for the coefficient of quartile variation","volume":"48","author":"Altunkaynak","year":"2019","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"827","DOI":"10.1080\/03610918.2013.794288","article-title":"A parametric bootstrap approach for one-way ANOVA under unequal variances with unbalanced data","volume":"44","author":"Zhang","year":"2015","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"108676","DOI":"10.1016\/j.ress.2022.108676","article-title":"Probabilistic forecasting informed failure prognostics framework for improved RUL prediction under uncertainty: A transformer case study","volume":"226","author":"Aizpurua","year":"2022","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1294","DOI":"10.1016\/j.jspi.2011.12.006","article-title":"Bayesian confidence intervals for means and variances of lognormal and bivariate lognormal distributions","volume":"142","author":"Harvey","year":"2012","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"526","DOI":"10.22237\/jmasm\/1478003400","article-title":"Bayesian inference for median of the lognormal distribution","volume":"15","author":"Rao","year":"2016","journal-title":"J. Mod. Appl. Stat. Methods"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/j.jkss.2018.03.002","article-title":"Bayesian methods for dealing with missing data problems","volume":"47","author":"Ma","year":"2018","journal-title":"J. Korean Stat. Soc."},{"key":"ref_24","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_25","unstructured":"Casella, G., and Berger, R.L. (2002). Statistical Inference, Duxbury."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1944\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T09:29:31Z","timestamp":1763026171000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/11\/1944"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,13]]},"references-count":25,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["sym17111944"],"URL":"https:\/\/doi.org\/10.3390\/sym17111944","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,13]]}}}