{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:46:41Z","timestamp":1760244401037,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,17]],"date-time":"2022-12-17T00:00:00Z","timestamp":1671235200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"The Deputyship for Research &amp; Innovation, Ministry of Education in Saudi Arabia","award":["INST 053"],"award-info":[{"award-number":["INST 053"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A new four-parameter lifetime distribution called the beta binomial exponential 2 (BBE2) distribution is proposed. Some mathematical features, including quantile function, moments, generating function and characteristic function, of the BBE2 distribution, are computed. When the life test is truncated at a predetermined time, acceptance sampling plans (ASP) are constructed for the BBE2 distribution. The truncation time is supposed to represent the median lifetime of the BBE2 distribution with predetermined factors for the smallest sample size required to guarantee that the prescribed life test is achieved at a given consumer\u2019s risk. Some numerical results for a given consumer\u2019s risk, BBE2 distribution parameters and truncation time are derived. Classical (maximum likelihood and maximum product of spacing estimation methods) and Bayesian estimation approaches are utilized to estimate the model parameters. The performance of the model parameters is examined through the simulation study by using the three different approaches of estimation. Subsequently, we examine real-world data applications to demonstrate the versatility and potential of the BBE2 model. A real-world application demonstrates that the new distribution can offer a better fit than other competitive lifetime models.<\/jats:p>","DOI":"10.3390\/axioms11120740","type":"journal-article","created":{"date-parts":[[2022,12,19]],"date-time":"2022-12-19T08:41:41Z","timestamp":1671439301000},"page":"740","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Statistical Inference of the Beta Binomial Exponential 2 Distribution with Application to Environmental Data"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9226-3267","authenticated-orcid":false,"given":"Osama H. Mahmoud","family":"Hassan","sequence":"first","affiliation":[{"name":"Department of Quantitative Methods, School of Business, King Faisal University, Al Hofuf 31982, Al-Ahsa, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abdullah H.","family":"Al-Nefaie","sequence":"additional","affiliation":[{"name":"Department of Quantitative Methods, School of Business, King Faisal University, Al Hofuf 31982, Al-Ahsa, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ahmed R.","family":"El-Saeed","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Obour High Institute for Management & Informatics, Obour 11848, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Shaked, M., and Shanthikumar, J.G. (2007). Stochastic Orders, Springer.","DOI":"10.1007\/978-0-387-34675-5"},{"key":"ref_2","unstructured":"Barlow, R.E., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing, Holt, Rinehart, and Winston."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"5392","DOI":"10.1016\/j.apm.2014.04.028","article-title":"A lifetime model with increasing failure rate","volume":"38","author":"Bakouch","year":"2014","journal-title":"Appl. Math. Modell."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1002\/bimj.4710300714","article-title":"On generalized log-logistic model for censored survival data","volume":"30","author":"Singh","year":"1998","journal-title":"Biom. J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1081\/STA-120003130","article-title":"Beta-normal distribution and its applications","volume":"31","author":"Eugene","year":"2002","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1155\/S1024123X04403068","article-title":"The beta Gumbel distribution","volume":"10","author":"Nadarajah","year":"2004","journal-title":"Math. Probl. Eng."},{"key":"ref_7","first-page":"15","article-title":"The beta Frechet distribution","volume":"14","author":"Nadarajah","year":"2004","journal-title":"Far East J. Theor. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"173","DOI":"10.22237\/jmasm\/1177992960","article-title":"Beta-Weibull Distribution: Some Properties and Applications to Censored Data","volume":"6","author":"Lee","year":"2007","journal-title":"J. Modern Appl. Statist. Methods"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1080\/00949655.2011.603089","article-title":"The beta Weibull-geometric distribution","volume":"83","author":"Bidram","year":"2013","journal-title":"J. Statist. Comput. Simul."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1080\/00949650802552402","article-title":"The beta generalized exponential distribution","volume":"80","author":"Santos","year":"2010","journal-title":"J. Statist. Comput. Simul."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1007\/s10985-010-9161-1","article-title":"The beta modified Weibull distribution","volume":"16","author":"Silva","year":"2010","journal-title":"Lifetime Data Anal."},{"key":"ref_12","first-page":"113","article-title":"The beta inverse Weibull distribution","volume":"3","author":"Khan","year":"2010","journal-title":"Int. Trans. Math. Sci. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2414","DOI":"10.1016\/j.matcom.2011.03.006","article-title":"The Beta Generalized Pareto distribution with application to lifetime data","volume":"81","author":"Mahmoudi","year":"2011","journal-title":"Math. Compu. Simul."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1080\/00949655.2011.615838","article-title":"The beta exponentiated Weibull distribution","volume":"83","author":"Cordeiro","year":"2013","journal-title":"J. Statist. Comput. Simul."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"189","DOI":"10.18576\/jsap\/070117","article-title":"The odd Fr\u00e9chet- G family of probability distributions","volume":"7","author":"Haq","year":"2018","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.6339\/JDS.2013.11(1).1086","article-title":"The exponentiated generalized class of distributions","volume":"11","author":"Cordeiro","year":"2013","journal-title":"J. Data Sci."},{"key":"ref_17","first-page":"53","article-title":"Odd generalized N-H generated family of distributions with application to exponential model","volume":"16","author":"Zubair","year":"2020","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s40300-013-0007-y","article-title":"A new method for generating families of continuous distributions","volume":"71","author":"Alzaatreh","year":"2013","journal-title":"Metron"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Aldahlan, M.A., Jamal, F., Chesneau, C., Elbatal, I., and Elgarhy, M. (2020). Exponentiated power generalized Weibull power series family of distributions: Properties, estimation and applications. PLoS ONE, 15.","DOI":"10.1371\/journal.pone.0230004"},{"key":"ref_20","first-page":"1253","article-title":"The Weibull-G family of probability distributions","volume":"12","author":"Bourguignon","year":"2014","journal-title":"J. Data Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"245","DOI":"10.18187\/pjsor.v13i2.1560","article-title":"Type II half Logistic family of distributions with applications","volume":"13","author":"Hassan","year":"2017","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Alotaibi, N., Elbatal, I., Almetwally, E.M., Alyami, S.A., Al-Moisheer, A.S., and Elgarhy, M. (2022). Truncated Cauchy Power Weibull-G Class of Distributions: Bayesian and Non-Bayesian Inference Modelling for COVID-19 and Carbon Fiber Data. Mathematics, 10.","DOI":"10.3390\/math10091565"},{"key":"ref_23","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_24","doi-asserted-by":"crossref","unstructured":"Algarni, A., MAlmarashi, A., Elbatal, I., SHassan, A., Almetwally, E.M., MDaghistani, A., and Elgarhy, M. (2021). Type I half logistic Burr XG family: Properties, Bayesian, and non-Bayesian estimation under censored samples and applications to COVID-19 data. Math. Probl. Eng.","DOI":"10.1155\/2021\/5461130"},{"key":"ref_25","first-page":"74","article-title":"Sine Topp-Leone-G family of distributions: Theory and applications","volume":"18","author":"Elbatal","year":"2020","journal-title":"Open Phys."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Bantan, R.A., Jamal, F., Chesneau, C., and Elgarhy, M. (2019). A New Power Topp\u2013Leone Generated Family of Distributions with Applications. Entropy, 21.","DOI":"10.3390\/e21121177"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Bantan, R.A., Jamal, F., Chesneau, C., and Elgarhy, M. (2019). Truncated inverted Kumaraswamy generated family of distributions with applications. Entropy, 21.","DOI":"10.3390\/e21111089"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"6543","DOI":"10.1080\/03610926.2015.1130839","article-title":"A new method for generating distributions with an application to exponential distribution","volume":"46","author":"Mahdavi","year":"2017","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Bantan, R.A., Chesneau, C., Jamal, F., and Elgarhy, M. (2020). On the Analysis of New COVID-19 Cases in Pakistan Using an Exponentiated Version of the M Family of Distributions. Mathematics, 8.","DOI":"10.3390\/math8060953"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Badr, M.M., Elbatal, I., Jamal, F., Chesneau, C., and Elgarhy, M. (2020). The transmuted odd Fr\u00e9chet-G family of distributions: Theory and applications. Mathematics, 8.","DOI":"10.3390\/math8060958"},{"key":"ref_31","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2007). Table of Integrals, Series, and Products, Academic Press."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2368","DOI":"10.1080\/02664763.2016.1254729","article-title":"A generalized binomial exponential 2 distribution. Modeling and applications to hydrologic events","volume":"44","author":"Asgharzadeh","year":"2017","journal-title":"J. Appl. Stat."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1016\/j.ress.2005.05.008","article-title":"The beta exponential distribution","volume":"91","author":"Nadarajah","year":"2006","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"187","DOI":"10.22237\/jmasm\/1177993020","article-title":"On the Properties of Beta-Gamma Distribution","volume":"1","author":"Kong","year":"2007","journal-title":"J. Modern Appl. Statist. Methods"},{"key":"ref_35","first-page":"991","article-title":"Power Inverted Topp-Leone in Acceptance Sampling Plans","volume":"67","author":"Abushal","year":"2021","journal-title":"Comput. Mater. Contin."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Alyami, S.A., Elgarhy, M., Elbatal, I., Almetwally, E.M., Alotaibi, N., and El-Saeed, A.R. (2022). Fr\u00e9chet Binomial Distribution: Statistical Properties, Acceptance Sampling Plan, Statistical Inference and Applications to Lifetime Data. Axioms, 11.","DOI":"10.3390\/axioms11080389"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"13628","DOI":"10.3934\/mbe.2022636","article-title":"Acceptance sampling plans for the three-parameter inverted Topp\u2013Leone model","volume":"19","author":"Nassr","year":"2022","journal-title":"Math. Biosci. Eng."},{"key":"ref_38","first-page":"169","article-title":"Acceptance sampling plans for inverse Weibull distribution based on truncated life test. Life Cycle Reliab","volume":"6","author":"Singh","year":"2017","journal-title":"Saf. Eng."},{"key":"ref_39","unstructured":"Varian, H.R. (2000). A Bayesian approach to real estate assessment. Variants in Economic Theory: Selected Works of H. R.Varian, Edward Elgar Publishing. Chapter 9."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1393","DOI":"10.1080\/00949655.2010.486762","article-title":"Bayesian analysis for the two-parameter Pareto distribution based on record values and times","volume":"81","author":"Doostparast","year":"2011","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1080\/03610929608831715","article-title":"Point estimation under asymmetric loss functions for left-truncated exponential samples","volume":"25","author":"Calabria","year":"1996","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"143","DOI":"10.3758\/s13423-016-1015-8","article-title":"A simple introduction to Markov Chain Monte-Carlo sampling","volume":"25","author":"Cassey","year":"2018","journal-title":"Psychon. Bull. Rev."},{"key":"ref_43","unstructured":"Andrews, D.F., and Herzberg, A.M. (1985). Data: A Collection of Problems from Many Fields for the Student and Research Worker, Springer."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1323","DOI":"10.1080\/03610920701826088","article-title":"A generalization of the half-normal distribution with applications to life time data","volume":"37","author":"Cooray","year":"2008","journal-title":"Comm. Statist. Theory Methods"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"e01144","DOI":"10.1016\/j.heliyon.2019.e01144","article-title":"Kumaraswamy log-logistic Weibull distribution: Model, theory and application to lifetime and survival data","volume":"5","author":"Mdlongwa","year":"2019","journal-title":"Heliyon"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/740\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:43:11Z","timestamp":1760146991000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/740"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,17]]},"references-count":45,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["axioms11120740"],"URL":"https:\/\/doi.org\/10.3390\/axioms11120740","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,12,17]]}}}