{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T16:33:07Z","timestamp":1780417987241,"version":"3.54.1"},"reference-count":42,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,8]],"date-time":"2022-08-08T00:00:00Z","timestamp":1659916800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002713","name":"Imam Mohammad Ibn Saud Islamic University","doi-asserted-by":"publisher","award":["RG-21-09-15"],"award-info":[{"award-number":["RG-21-09-15"]}],"id":[{"id":"10.13039\/501100002713","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A new class of distribution called the Fr\u00e9chet binomial (FB) distribution is proposed. The new suggested model is very flexible because its probability density function can be unimodal, decreasing and skewed to the right. Furthermore, the hazard rate function can be increasing, decreasing, up-side-down and reversed-J form. Important mixture representations of the probability density function (pdf) and cumulative distribution function (cdf) are computed. Numerous sub-models of the FB distribution are explored. Numerous statistical and mathematical features of the FB distribution such as the quantile function (QUNF); moments (MO); incomplete MO (IMO); conditional MO (CMO); MO generating function (MOGF); probability weighted MO (PWMO); order statistics; and entropy are computed. When the life test is shortened at a certain time, acceptance sampling (ACS) plans for the new proposed distribution, FB distribution, are produced. The truncation time is supposed to be the median lifetime of the FB distribution multiplied by a set of parameters. The smallest sample size required ensures that the specified life test is obtained at a particular consumer\u2019s risk. The numerical results for a particular consumer\u2019s risk, FB distribution parameters and truncation time are generated. We discuss the method of maximum likelihood to estimate the model parameters. A simulation study was performed to assess the behavior of the estimates. Three real datasets are used to illustrate the importance and flexibility of the proposed model.<\/jats:p>","DOI":"10.3390\/axioms11080389","type":"journal-article","created":{"date-parts":[[2022,8,8]],"date-time":"2022-08-08T21:09:18Z","timestamp":1659992958000},"page":"389","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Fr\u00e9chet Binomial Distribution: Statistical Properties, Acceptance Sampling Plan, Statistical Inference and Applications to Lifetime Data"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5507-9399","authenticated-orcid":false,"given":"Salem A.","family":"Alyami","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1333-3862","authenticated-orcid":false,"given":"Mohammed","family":"Elgarhy","sequence":"additional","affiliation":[{"name":"The Higher Institute of Commercial Sciences, Al Mahalla Al Kubra 31951, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ibrahim","family":"Elbatal","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia"},{"name":"Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3888-1275","authenticated-orcid":false,"given":"Ehab M.","family":"Almetwally","sequence":"additional","affiliation":[{"name":"Faculty of Business Administration, Delta University of Science and Technology, Gamasa 11152, Egypt"},{"name":"The Scientific Association for Studies and Applied Research, Al Manzalah 35646, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Naif","family":"Alotaibi","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ahmed R.","family":"El-Saeed","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Obour High Institute for Management & Informatics, Obour 11848, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,8]]},"reference":[{"key":"ref_1","first-page":"5","article-title":"Sur la loi des erreurs dobservation","volume":"33","year":"1924","journal-title":"Bull. Soc. Math. Moscou"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Kotz, S., and Nadarajah, S. (2000). Extreme Value Distributions: Theory and Applications, Imperial College Press.","DOI":"10.1142\/9781860944024"},{"key":"ref_3","first-page":"1","article-title":"The exponentiated Fr\u00e9chet distribution","volume":"14","author":"Nadarajah","year":"2003","journal-title":"Inter. Stat. Electron. J."},{"key":"ref_4","first-page":"15","article-title":"The beta Fr\u00e9chet distribution","volume":"14","author":"Nadarajah","year":"2004","journal-title":"Far East J. Theor. Stat."},{"key":"ref_5","first-page":"5553","article-title":"On the transmuted Fr\u00e9chet distribution","volume":"9","author":"Mahmoud","year":"2013","journal-title":"J. Appl. Sci. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4091","DOI":"10.1080\/03610926.2011.648785","article-title":"The Marshall\u2013Olkin Fr\u00e9chet distribution","volume":"42","author":"Krishna","year":"2013","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Alotaibi, R., Almetwally, E.M., Ghosh, I., and Rezk, H. (2022). Classical and Bayesian Inference on Finite Mixture of Exponentiated Kumaraswamy Gompertz and Exponentiated Kumaraswamy Fr\u00e9chet Distributions under Progressive Type II Censoring with Applications. Mathematics, 10.","DOI":"10.3390\/math10091496"},{"key":"ref_8","first-page":"379","article-title":"Transmuted exponentiated Fr\u00e9chet distribution: Properties and applications","volume":"3","author":"Elbatal","year":"2014","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"41","DOI":"10.17713\/ajs.v46i1.144","article-title":"The beta exponential Fr\u00e9chet distribution with applications","volume":"46","author":"Mead","year":"2016","journal-title":"Austrian J. Stat."},{"key":"ref_10","first-page":"294","article-title":"A note on Kumaraswamy Fr\u00e9chet distribution","volume":"8","author":"Mead","year":"2014","journal-title":"Australia"},{"key":"ref_11","first-page":"23","article-title":"The burr x Fr\u00e9chet distribution with its properties and applications","volume":"13","author":"Abouelmagd","year":"2018","journal-title":"Appl. Probab. Stat."},{"key":"ref_12","first-page":"1","article-title":"On a bivariate Fr\u00e9chet distribution","volume":"9","author":"Almetwally","year":"2020","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2167670","DOI":"10.1155\/2021\/2167670","article-title":"Statistical inference under censored data for the new exponential-X Fr\u00e9chet distribution: Simulation and application to leukemia data","volume":"2021","author":"Alzeley","year":"2021","journal-title":"Comput. Intell. Neurosci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2608","DOI":"10.1080\/02664763.2016.1142945","article-title":"The Weibull Fr\u00e9chet distribution and its applications","volume":"43","author":"Afify","year":"2016","journal-title":"J. Appl. Stat."},{"key":"ref_15","first-page":"41","article-title":"The Kumaraswamy Marshall-Olkin Fr\u00e9chet distribution: Properties and Applications","volume":"2","author":"Afify","year":"2016","journal-title":"J. ISOSS"},{"key":"ref_16","first-page":"129","article-title":"Some theoretical and computational aspects of the odd Lindley Fr\u00e9chet distribution","volume":"2","author":"Korkmaz","year":"2017","journal-title":"J. Stat. Stat. Actuar. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"129","DOI":"10.18576\/amis\/130117","article-title":"Alpha power transformed Fr\u00e9chet distribution","volume":"13","author":"Nasiru","year":"2019","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_18","first-page":"170","article-title":"The alpha power Weibull Fr\u00e9chet distribution: Properties and applications","volume":"5","author":"Eghwerido","year":"2020","journal-title":"Turk. J. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1166\/jctn.2019.7734","article-title":"Truncated Weibull Fr\u00e9chet Distribution: Statistical Inference and Applications","volume":"16","author":"Hassan","year":"2019","journal-title":"J. Comput. Theor. Nanosci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"189","DOI":"10.18576\/jsap\/070117","article-title":"The odd Fr\u00e9chet-G class of probability distributions","volume":"7","author":"Haq","year":"2018","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Badr, M., Elbatal, I., Jamal, F., Chesneau, C., and Elgarhy, M. (2020). The Transmuted Odd Fr\u00e9chet-G class of Distributions: Theory and Applications. Mathematics, 8.","DOI":"10.3390\/math8060958"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2931326","DOI":"10.1155\/2018\/2931326","article-title":"Extended Odd Fr\u00e9chet-G class of Distributions","volume":"2018","author":"Nasiru","year":"2018","journal-title":"J. Probab. Stat."},{"key":"ref_23","first-page":"267","article-title":"A new generalization of the Fr\u00e9chet distribution: Properties and application","volume":"79","author":"Jayakumar","year":"2019","journal-title":"Statistica"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Baharith, L., and Alamoudi, H. (2021). The exponentiated Fr\u00e9chet Generator of \u00b4 Distributions with Applications. Symmetry, 13.","DOI":"10.3390\/sym13040572"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Alyami, S.A., Babu, M.G., Elbatal, I., Alotaibi, N., and Elgarhy, M. (2022). Type II Half-Logistic Odd Fr\u00e9chet Class of Distributions: Statistical Theory and Applications. Symmetry, 14.","DOI":"10.3390\/sym14061222"},{"key":"ref_26","first-page":"791","article-title":"and Al-Marzouki, S. Generalized Truncated Fr\u00e9chet Generated Family Distributions an dTheir Applications","volume":"126","author":"ZeinEldin","year":"2021","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_27","first-page":"437","article-title":"Topp-Leone Odd Fr\u00e9chet Generated Family of Distributions with Applications to COVID-19 Data Sets","volume":"125","author":"Jamal","year":"2020","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"994","DOI":"10.1214\/aos\/1176350046","article-title":"Skewness and asymmetry: Measures and orderings","volume":"14","author":"MacGillivray","year":"1986","journal-title":"Ann. Stat."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Arnold, B.C., Balakrishnan, N., and Nagraja, H.N. (2008). A First Course in Order Statistics, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9780898719062"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s41872-017-0022-8","article-title":"Acceptance sampling plans for inverse Weibull distribution based on truncated life test","volume":"6","author":"Singh","year":"2017","journal-title":"Life Cycle Reliab. Saf. Eng."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Elbatal, I., Alotaibi, N., Almetwally, E.M., Alyami, S.A., and Elgarhy, M. (2022). On Odd Perks-G Class of Distributions: Properties, Regression Model, Discretization, Bayesian and Non-Bayesian Estimation, and Applications. Symmetry, 14.","DOI":"10.3390\/sym14050883"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s40064-016-3464-y","article-title":"The power Lomax distribution with an application to bladder cancer data","volume":"5","author":"Rady","year":"2016","journal-title":"SpringerPlus"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1080\/16583655.2019.1588488","article-title":"Alpha power inverse Weibull distribution with reliability application","volume":"13","author":"Basheer","year":"2019","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1317","DOI":"10.1081\/STA-120021561","article-title":"On the exponentiated Weibull distribution","volume":"32","author":"Nassar","year":"2003","journal-title":"Commun. -Stat.-Theory Methods"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1080\/03610910701236081","article-title":"Failure data analysis with extended Weibull distribution","volume":"36","author":"Zhang","year":"2007","journal-title":"Commun. Stat. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s40745-021-00329-w","article-title":"The odd Weibull inverse topp\u2014leone distribution with applications to COVID-19 data","volume":"9","author":"Almetwally","year":"2022","journal-title":"Ann. Data Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"137","DOI":"10.18187\/pjsor.v17i1.3402","article-title":"Marshall\u2014Olkin alpha power lomax distribution: Estimation methods, applications on physics and economics","volume":"17","author":"Almongy","year":"2021","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1855","DOI":"10.1080\/03610920601126571","article-title":"Marshall\u2013Olkin extended Lomax distribution and its application to censored data","volume":"36","author":"Ghitany","year":"2007","journal-title":"Commun. Stat. Methods"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"82","DOI":"10.22606\/jaam.2019.42006","article-title":"New generalization of length biased exponential distribution with applications","volume":"4","author":"Maxwell","year":"2019","journal-title":"J. Adv. Appl. Math."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Jorgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution, Springer.","DOI":"10.1007\/978-1-4612-5698-4"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"163","DOI":"10.4236\/ojs.2020.102013","article-title":"Location based generalized Akash distribution: Properties and applications","volume":"10","author":"Tharshan","year":"2020","journal-title":"Open J. Stat."},{"key":"ref_42","unstructured":"Gross, A.J., and Clark, V. (1975). Survival Distributions: Reliability Applications in the Biomedical Sciences, John Wiley & Sons."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/389\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:05:51Z","timestamp":1760141151000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/8\/389"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,8]]},"references-count":42,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["axioms11080389"],"URL":"https:\/\/doi.org\/10.3390\/axioms11080389","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,8,8]]}}}