{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T18:50:38Z","timestamp":1773514238580,"version":"3.50.1"},"reference-count":53,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,8,21]],"date-time":"2021-08-21T00:00:00Z","timestamp":1629504000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000956","name":"DSR","doi-asserted-by":"publisher","award":["D-229-150-1440"],"award-info":[{"award-number":["D-229-150-1440"]}],"id":[{"id":"10.13039\/501100000956","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this article, the \u201ctruncated-composed\u201d scheme was applied to the Burr X distribution to motivate a new family of univariate continuous-type distributions, called the truncated Burr X generated family. It is mathematically simple and provides more modeling freedom for any parental distribution. Additional functionality is conferred on the probability density and hazard rate functions, improving their peak, asymmetry, tail, and flatness levels. These characteristics are represented analytically and graphically with three special distributions of the family derived from the exponential, Rayleigh, and Lindley distributions. Subsequently, we conducted asymptotic, first-order stochastic dominance, series expansion, Tsallis entropy, and moment studies. Useful risk measures were also investigated. The remainder of the study was devoted to the statistical use of the associated models. In particular, we developed an adapted maximum likelihood methodology aiming to efficiently estimate the model parameters. The special distribution extending the exponential distribution was applied as a statistical model to fit two sets of actuarial and financial data. It performed better than a wide variety of selected competing non-nested models. Numerical applications for risk measures are also given.<\/jats:p>","DOI":"10.3390\/e23081088","type":"journal-article","created":{"date-parts":[[2021,8,22]],"date-time":"2021-08-22T21:47:52Z","timestamp":1629668872000},"page":"1088","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":48,"title":["The Truncated Burr X-G Family of Distributions: Properties and Applications to Actuarial and Financial Data"],"prefix":"10.3390","volume":"23","author":[{"given":"Rashad A. R.","family":"Bantan","sequence":"first","affiliation":[{"name":"Department of Marine Geology, Faculty of Marine Science, King AbdulAziz University, Jeddah 21551, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christophe","family":"Chesneau","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Universit\u00e9 de Caen, LMNO, Campus II, Science 3, 14032 Caen, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6192-9890","authenticated-orcid":false,"given":"Farrukh","family":"Jamal","sequence":"additional","affiliation":[{"name":"Department of Statistics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan"}],"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"}]},{"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, Algarbia 31951, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,8,21]]},"reference":[{"key":"ref_1","first-page":"897","article-title":"Method for generating distributions and families of probability distributions: The univariate case","volume":"48","author":"Brito","year":"2019","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Cordeiro, G.M., Silva, R.B., and Nascimento, A.D.C. (2020). Recent Advances in Lifetime and Reliability Models, Bentham Sciences Publishers.","DOI":"10.2174\/97816810834521200101"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s40488-016-0052-1","article-title":"Compounding of distributions: A survey and new generalized classes","volume":"3","author":"Tahir","year":"2016","journal-title":"J. Stat. Distrib. Appl."},{"key":"ref_4","first-page":"51","article-title":"[0,1] truncated Fr\u00e9chet-G generator of distributions","volume":"7","author":"Abid","year":"2017","journal-title":"Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.5539\/ijsp.v6n5p1","article-title":"Truncated Weibull-G more flexible and more reliable than geta-G distribution","volume":"6","author":"Najarzadegan","year":"2017","journal-title":"Int. J. Stat. Probab."},{"key":"ref_6","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_7","first-page":"221","article-title":"Type II Fr\u00e9chet generated family of distributions","volume":"7","author":"Aldahlan","year":"2019","journal-title":"Int. J. Math. Its Appl."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Al-Babtain, A.A., Elbatal, I., Chesneau, C., and Jamal, F. (2020). The transmuted Muth generated class of distributions with applications. Symmetry, 12.","DOI":"10.3390\/sym12101677"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Al-Babtain, A.A., Elbatal, I., Chesneau, C., and Jamal, F. (2020). Box-Cox gamma-G family of distributions: Theory and applications. Mathematics, 8.","DOI":"10.3390\/math8101801"},{"key":"ref_10","first-page":"437","article-title":"Topp-Leone odd Fr\u00e9chet generated family of distributions with applications to Covid-19 datasets. CMES-Comput","volume":"125","author":"Jamal","year":"2020","journal-title":"Modeling Eng. Sci."},{"key":"ref_11","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_12","first-page":"593","article-title":"The xgamma family: Censored regression modelling and applications","volume":"18","author":"Cordeiro","year":"2020","journal-title":"REVSTAT-Stat. J."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Aldahlan, M.A., Jamal, F., Chesneau, C., Elgarhy, M., and Elbatal, I. (2020). The truncated Cauchy power family of distributions with inference and applications. Entropy, 22.","DOI":"10.3390\/e22030346"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Almarashi, A.M., Elgarhy, M., Jamal, F., and Chesneau, C. (2020). The exponentiated truncated inverse Weibull generated family of distributions with applications. Symmetry, 12.","DOI":"10.3390\/sym12040650"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Badr, M.A., 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_16","first-page":"34","article-title":"The Marshall-Olkin exponentiated generalized G family of distributions: Properties, applications and characterizations","volume":"13","author":"Yousof","year":"2020","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Bantan, R.A.R., 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_18","doi-asserted-by":"crossref","unstructured":"Bantan, R.A.R., Jamal, F., Chesneau, C., and Elgarhy, M. (2020). Type II Power Topp-Leone generated family of distributions with statistical inference and applications. Symmetry, 12.","DOI":"10.3390\/sym12010075"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1080\/16583655.2020.1732642","article-title":"A modified T-X family of distributions: Classical and Bayesian analysis","volume":"14","author":"Aslam","year":"2020","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1007\/s41096-020-00075-x","article-title":"Modified odd Weibull family of distributions: Properties and applications","volume":"21","author":"Chesneau","year":"2020","journal-title":"J. Indian Soc. Probab. Stat."},{"key":"ref_21","unstructured":"Jamal, F., Bakouch, H.S., and Nasir, M.A. (2020). A truncated general-G class of distributions with application to truncated Burr-G family. Revstat.- Stat. J., in press."},{"key":"ref_22","first-page":"617","article-title":"Type II general inverse exponential family of distributions","volume":"23","author":"Jamal","year":"2020","journal-title":"J. Stat. Manag. Syst."},{"key":"ref_23","first-page":"3","article-title":"The odd generalized gamma-G family of distributions: Properties, regressions and applications","volume":"80","author":"Nasir","year":"2020","journal-title":"Statistica"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"791","DOI":"10.32604\/cmes.2021.012169","article-title":"Generalized truncated Fr\u00e9chet generated family distributions and their applications","volume":"126","author":"ZeinEldin","year":"2021","journal-title":"CMES-Comput. Modeling Eng. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1214\/aoms\/1177731607","article-title":"Cumulative frequency functions","volume":"13","author":"Burr","year":"1942","journal-title":"Ann. Math. Stat."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/S0378-3758(97)00038-4","article-title":"Empirical Bayes estimation of P(Y<X) and characterizations of Burr-type X model","volume":"64","author":"Ahmad","year":"1997","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2307","DOI":"10.1080\/03610929108830633","article-title":"Bayesian prediction bounds for the Burr type X model","volume":"20","year":"1991","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1080\/03610918.2012.714033","article-title":"Control charts for monitoring Burr type-X percentiles","volume":"43","author":"Lio","year":"2014","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/S0898-1221(98)00143-6","article-title":"Order statistics from the Burr type X model","volume":"36","author":"Raqab","year":"1998","journal-title":"Comput. Math. Appl."},{"key":"ref_30","first-page":"179","article-title":"Burr type X distribution: Revisited","volume":"4","author":"Raqab","year":"2006","journal-title":"J. Probab. Stat. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"3092","DOI":"10.1080\/00949655.2014.951359","article-title":"Higher order inference for stress-strength reliability with independent Burr-type X distributions","volume":"85","author":"Smith","year":"2015","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_32","first-page":"225","article-title":"Inference for P(Y<X) in the Burr type X model","volume":"7","author":"Surles","year":"1998","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1023\/A:1011352923990","article-title":"Inference for reliability and stress-strength for a scaled Burr Type X distribution","volume":"7","author":"Surles","year":"2001","journal-title":"Lifetime Data Anal."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/j.jspi.2003.10.003","article-title":"Some properties of a scaled Burr type X distribution","volume":"128","author":"Surles","year":"2005","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"288","DOI":"10.2991\/jsta.2017.16.3.2","article-title":"The Burr X generator of distributions for lifetime data","volume":"16","author":"Yousof","year":"2016","journal-title":"J. Stat. Theory Appl."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Korkmaz, M.C., Altun, E., Yousof, H.M., Afify, A.Z., and Nadarajah, S. (2018). The Burr X Pareto distribution: Properties, applications and VaR estimation. J. Risk Financ. Manag., 11.","DOI":"10.3390\/jrfm11010001"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1109\/TR.1987.5222310","article-title":"How to identify bathtub hazard rate","volume":"36","author":"Aarset","year":"1987","journal-title":"IEEE Trans. Reliab."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Gilchrist, W. (2000). Statistical Modelling with Quantile Functions, CRC Press.","DOI":"10.1201\/9781420035919"},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Shaked, M., and Shanthikumar, J.G. (2007). Stochastic Orders, Wiley.","DOI":"10.1007\/978-0-387-34675-5"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Amigo, J.M., Balogh, S.G., and Hernandez, S. (2018). A brief review of generalized entropies. Entropy, 20.","DOI":"10.3390\/e20110813"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1111\/1467-9965.00068","article-title":"Coherent measures of risk","volume":"9","author":"Artzner","year":"1999","journal-title":"Math. Financ."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Fischer, M., Moser, T., and Pfeuffer, M. (2018). A discussion on recent risk measures with application to credit risk: Calculating risk contributions and identifying risk concentrations. Risks, 6.","DOI":"10.3390\/risks6040142"},{"key":"ref_44","unstructured":"Casella, G., and Berger, R.L. (1990). Statistical Inference, Brooks\/Cole Publishing Company."},{"key":"ref_45","unstructured":"R Development Core Team (2005). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing. Available online: http:\/\/www.R-project.org."},{"key":"ref_46","first-page":"53","article-title":"A new generalization of Burr XII distribution","volume":"12","author":"Mead","year":"2014","journal-title":"J. Stat. Adv. Theory Appl."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1337","DOI":"10.1007\/s10614-018-9806-9","article-title":"Risk: An R package for financial risk measures","volume":"53","author":"Chan","year":"2019","journal-title":"Comput. Econ."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.jmva.2017.05.001","article-title":"Score tests for covariate effects in conditional copulas","volume":"159","author":"Gijbels","year":"2017","journal-title":"J. Multivar. Anal."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.insmatheco.2021.04.007","article-title":"Infinitely stochastic micro reserving","volume":"100","author":"Maciak","year":"2021","journal-title":"Insur. Math. Econ."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"28","DOI":"10.1016\/j.insmatheco.2014.02.007","article-title":"Conditional least squares and copulae in claims reserving for a single line of business","volume":"56","author":"Okhrin","year":"2014","journal-title":"Insur. Math. Econ."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"472","DOI":"10.1007\/s10891-012-0675-0","article-title":"Modeling of the damaged state by the finite-element method on simultaneous action of contact and noncontact loads","volume":"85","author":"Sherbakov","year":"2012","journal-title":"J. Eng. Phys. Thermophys."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Sherbakov, S. (2013). Spatial stress-strain state of tribofatigue system in roll-shaft contact zone. Strength Mater., 45.","DOI":"10.1007\/s11223-013-9430-9"},{"key":"ref_53","first-page":"207","article-title":"Measurement and Real Time Analysis of Local Damage in Wear-and-Fatigue Tests","volume":"10","author":"Sherbakov","year":"2019","journal-title":"Devices Methods Meas."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1088\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:48:47Z","timestamp":1760165327000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1088"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,8,21]]},"references-count":53,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["e23081088"],"URL":"https:\/\/doi.org\/10.3390\/e23081088","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,8,21]]}}}