{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,16]],"date-time":"2026-10-16T10:29:55Z","timestamp":1792146595212,"version":"4.4.0"},"reference-count":35,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,25]],"date-time":"2021-01-25T00:00:00Z","timestamp":1611532800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Fahad Uniersity of Petroleum and Minerals","award":["SB191042"],"award-info":[{"award-number":["SB191042"]}]},{"name":"Bangabandhu Sheikh Mujibur Rahman Science &amp; Technology University","award":["4829"],"award-info":[{"award-number":["4829"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>(1) Background: The literature discusses the inverse Maxwell distribution theoretically without application. Control charting is promising, but needs development for inverse Maxwell processes. (2) Methods: Thus, we develop the VIM control chart for monitoring the inverse Maxwell scale parameter and studied its statistical properties. The chart\u2019s performance is evaluated using power curves and run length properties. (3) Results: Further, we use simulated data to compare the shift detection capability of our chart with Weibull, gamma, and lognormal charts. (4) Conclusion: The analysis demonstrates our chart\u2019s efficiency for monitoring skewed processes. Finally, we apply our chart for monitoring real world lifetimes of car brake pads.<\/jats:p>","DOI":"10.3390\/sym13020189","type":"journal-article","created":{"date-parts":[[2021,1,25]],"date-time":"2021-01-25T12:28:31Z","timestamp":1611577711000},"page":"189","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Inverse Maxwell Distribution and Statistical Process Control: An Efficient Approach for Monitoring Positively Skewed Process"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5333-2883","authenticated-orcid":false,"given":"M. Hafidz","family":"Omar","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals (KFUPM), Dhahran 31261, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sheikh Y.","family":"Arafat","sequence":"additional","affiliation":[{"name":"Department of Business Administration, Bangladesh Army International University of Science and Technology, Cumilla Cantonment 3501, Bangladesh"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5247-4820","authenticated-orcid":false,"given":"M. Pear","family":"Hossain","sequence":"additional","affiliation":[{"name":"Department of Statistics, Bangabandhu Sheikh Mujibur Rahman Science and Technology University, Gopalganj 8100, Bangladesh"},{"name":"Department of Biomedical Sciences, Jockey Club College of Veterinary Medicine and Life Sciences, City University of Hong Kong, Kowloon, Hong Kong"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Muhammad","family":"Riaz","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals (KFUPM), Dhahran 31261, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,25]]},"reference":[{"key":"ref_1","first-page":"32","article-title":"Inverse gaussian distribution and its application","volume":"77","author":"Sato","year":"1994","journal-title":"Electron. Commun. Jpn. (Part III Fundam. Electron. Sci.)"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1080\/00949655.2010.550291","article-title":"Reliability estimation in Maxwell distribution with progressively Type-II censored data","volume":"82","author":"Krishna","year":"2012","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1080\/00949655.2013.822871","article-title":"Estimation procedures for Maxwell distribution under type-I progressive hybrid censoring scheme","volume":"85","author":"Tomer","year":"2015","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2809","DOI":"10.1103\/PhysRevE.61.2809","article-title":"Deviation from Maxwell distribution in granular gases with constant restitution coefficient","volume":"61","author":"Brilliantov","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_5","first-page":"197","article-title":"On the Bayesian Estimation for two Component Mixture of Maxwell Distribution, Assuming Type I Censored Data","volume":"2","author":"Mohsin","year":"2012","journal-title":"Int. J. Appl. Sci. Technol."},{"key":"ref_6","first-page":"1","article-title":"Discrete maxwell distribution","volume":"2007","author":"Krishna","year":"2007","journal-title":"InterStat. J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"31","DOI":"10.2174\/1876527001103010031","article-title":"A Note on the Maximum Likelihood Estimators for the Mixture of Maxwell Distributions Using Type-I Censored Scheme","volume":"3","author":"Mohsin","year":"2011","journal-title":"Open Stat. Probab. J."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1016\/j.cie.2017.03.023","article-title":"Estimation of mixture Maxwell parameters and its possible industrial application","volume":"107","author":"Hossain","year":"2017","journal-title":"Comput. Ind. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s11222-008-9072-0","article-title":"Model-based clustering with non-elliptically contoured distributions","volume":"19","author":"Karlis","year":"2008","journal-title":"Stat. Comput."},{"key":"ref_10","first-page":"23","article-title":"Inverse Maxwell Distribution as a Survival Model, Genesis and Parameter Estimation","volume":"2","author":"Singh","year":"2014","journal-title":"Res. J. Math. Stat. Sci."},{"key":"ref_11","first-page":"52","article-title":"Estimation of the Parameter in the Size-Biased Inverse Maxwell Distribution","volume":"10","author":"Singh","year":"2014","journal-title":"Int. J. Stat. Math."},{"key":"ref_12","first-page":"1835","article-title":"Bayesian Estimation of Parameter of Inverse Maxwell Distribution via Size-Biased Sampling","volume":"3","author":"Singh","year":"2014","journal-title":"Int. J. Sci. Res."},{"key":"ref_13","first-page":"13","article-title":"Bayes Estimation of Parameter in Inverse Maxwell Distribution under Weighted Quadratic Loss Function","volume":"4","author":"Loganathan","year":"2017","journal-title":"Int. J. Sci. Res. Math. Stat. Sci."},{"key":"ref_14","unstructured":"(2020, April 04). NIST\/SEMATECH e-Handbook of Statistical Methods, Available online: https:\/\/www.itl.nist.gov\/div898\/handbook\/."},{"key":"ref_15","unstructured":"Montgomery, D.C. (2009). Introduction to Statistical Quality Control, John Wiley & Sons. [6th ed.]."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1080\/00224065.2000.11980013","article-title":"Controversies and Contradictions in Statistical Process Control","volume":"32","author":"Woodall","year":"2000","journal-title":"J. Qual. Technol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1080\/00401706.1959.10489860","article-title":"Control Chart Tests Based on Geometric Moving Averages","volume":"1","author":"Roberts","year":"1959","journal-title":"Technometrics"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1093\/biomet\/41.1-2.100","article-title":"Continuous Inspection Schemes","volume":"41","author":"Page","year":"1954","journal-title":"Biometrika"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1080\/00949655.2016.1222391","article-title":"New V control chart for the Maxwell distribution","volume":"87","author":"Hossain","year":"2017","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1923","DOI":"10.1007\/s00170-018-2679-1","article-title":"On designing Maxwell CUSUM control chart: An efficient way to monitor failure rates in boring processes","volume":"100","author":"Hossain","year":"2018","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5649","DOI":"10.1080\/00207540701325082","article-title":"A control chart for the Gamma distribution as a model of time between events","volume":"45","author":"Zhang","year":"2007","journal-title":"Int. J. Prod. Res."},{"key":"ref_22","first-page":"210","article-title":"Shewhart Control Charts for Rayleigh Distribution in the Presence of Type I Censored Data","volume":"2","author":"Raza","year":"2016","journal-title":"J. ISOSS"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1675","DOI":"10.1002\/qre.2354","article-title":"DEWMA control charts for censored data using Rayleigh lifetimes","volume":"34","author":"Raza","year":"2018","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/16843703.2017.1304044","article-title":"A control chart for the lognormal standard deviation","volume":"15","author":"Huang","year":"2017","journal-title":"Qual. Technol. Quant. Manag."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Arafat, S.Y., Hossain, M.P., Ajadi, J.O., and Riaz, M. (2019). On the Development of EWMA Control Chart for Inverse Maxwell Distribution. J. Test. Eval., 49.","DOI":"10.1520\/JTE20190082"},{"key":"ref_26","first-page":"1","article-title":"A Review on Inverse Maxwell Distribution with Its Statistical Properties and Applications","volume":"14","author":"Tomer","year":"2020","journal-title":"J. Stat. Theory Pr."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Klugman, S.A., Panjer, H.H., and Willmot, G.E. (2012). Loss Models from Data to Decisions, John Wiley & Sons. [4th ed.].","DOI":"10.1002\/9781118787106"},{"key":"ref_28","first-page":"95","article-title":"Control charts for skewed distributions: Weibull, gamma, and lognormal","volume":"9","author":"Derya","year":"2012","journal-title":"Metod. Zv."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1790","DOI":"10.3923\/jas.2007.1790.1794","article-title":"On Efficient Confidence Intervals for the Log-Normal Mean","volume":"7","author":"Chami","year":"2007","journal-title":"J. Appl. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1407","DOI":"10.1002\/qre.1841","article-title":"Control Charts for the Lognormal Mean","volume":"32","author":"Huang","year":"2015","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1080\/00401706.1968.10490608","article-title":"A Chart for Sequentially Testing Observed Arithmetic Means from Lognormal Populations against a Given Standard","volume":"1","author":"Joffe","year":"1968","journal-title":"Technometrics"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1691","DOI":"10.1002\/qre.1702","article-title":"EWMA Dispersion Control Charts for Normal and Non-normal Processes","volume":"31","author":"Abbasi","year":"2015","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Lawless, J.F. (2002). Statistical Models and Methods for Lifetime Data, Wiley-Interscience.","DOI":"10.1002\/9781118033005"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"465","DOI":"10.18187\/pjsor.v11i4.1008","article-title":"Length-biased Weighted Maxwell Distribution","volume":"11","author":"Modi","year":"2015","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Mathew, J., and Chesneau, C. (2020). Marshall\u2013Olkin Length-Biased Maxwell Distribution and Its Applications. Math. Comput. 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