{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T15:51:47Z","timestamp":1770997907877,"version":"3.50.1"},"reference-count":37,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,8]],"date-time":"2021-07-08T00:00:00Z","timestamp":1625702400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Specifying a proper statistical model to represent asymmetric lifetime data with high kurtosis is an open problem. In this paper, the three-parameter, modified, slashed, generalized Rayleigh family of distributions is proposed. Its structural properties are studied: stochastic representation, probability density function, hazard rate function, moments and estimation of parameters via maximum likelihood methods. As merits of our proposal, we highlight as particular cases a plethora of lifetime models, such as Rayleigh, Maxwell, half-normal and chi-square, among others, which are able to accommodate heavy tails. A simulation study and applications to real data sets are included to illustrate the use of our results.<\/jats:p>","DOI":"10.3390\/sym13071226","type":"journal-article","created":{"date-parts":[[2021,7,8]],"date-time":"2021-07-08T10:42:17Z","timestamp":1625740937000},"page":"1226","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["A Generalized Rayleigh Family of Distributions Based on the Modified Slash Model"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9064-2597","authenticated-orcid":false,"given":"Inmaculada","family":"Barranco-Chamorro","sequence":"first","affiliation":[{"name":"Departamento de Estad\u00edstica e Investigaci\u00f3n Operativa, Facultad de Matem\u00e1ticas, Universidad de Sevilla, 41012 Sevilla, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9847-1497","authenticated-orcid":false,"given":"Yuri A.","family":"Iriarte","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Facultad de Ciencias B\u00e1sicas, Universidad de Antofagasta,  Antofagasta 1240000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8092-9666","authenticated-orcid":false,"given":"Yolanda M.","family":"G\u00f3mez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Facultad de Ingenier\u00eda, Universidad de Atacama,  Copiap\u00f3 1530000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juan M.","family":"Astorga","sequence":"additional","affiliation":[{"name":"Departamento de Tecnolog\u00edas de la Energ\u00eda, Facultad Tecnol\u00f3gica, Universidad de Atacama,  Copiap\u00f3 1530000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3726-5507","authenticated-orcid":false,"given":"H\u00e9ctor W.","family":"G\u00f3mez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Facultad de Ciencias B\u00e1sicas, Universidad de Antofagasta,  Antofagasta 1240000, Chile"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,8]]},"reference":[{"key":"ref_1","first-page":"395","article-title":"Inferential procedures on a generalized Rayleigh variate I","volume":"21","year":"1976","journal-title":"Apl. Mat."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"167","DOI":"10.6028\/jres.066D.020","article-title":"Some problems connected with Rayleigh distributions","volume":"66","author":"Siddiqui","year":"1962","journal-title":"J. Res. Natl. Bur. Stand. Sect. D Radio Propag."},{"key":"ref_3","unstructured":"Kotz, S., Johnson, N.L., and Read, C.B. (1986). Rayleigh distribution. Encyclopedia of Statistical Sciences, Wiley."},{"key":"ref_4","unstructured":"Abramowitz, M., and Stegun, I.E. (1970). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables."},{"key":"ref_5","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2000). Table of Integrals, Series, and Products, Academic Press."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1016\/j.spl.2006.11.006","article-title":"A new family of slash-distributions with elliptical contours","volume":"77","author":"Quintana","year":"2007","journal-title":"Stat. Probab. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2025","DOI":"10.1080\/03610926.2019.1568484","article-title":"Generalized modified slash distribution with applications","volume":"49","author":"Reyes","year":"2020","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4686","DOI":"10.1080\/03610926.2015.1066811","article-title":"Slashed generalized Rayleigh distribution","volume":"46","author":"Iriarte","year":"2017","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1007\/s00362-011-0391-4","article-title":"An extension of the half-normal distribution","volume":"53","author":"Olmos","year":"2012","journal-title":"Stat. Pap."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"929","DOI":"10.1080\/02331888.2012.694441","article-title":"Modified slash distribution","volume":"47","author":"Reyes","year":"2013","journal-title":"Statistics"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1187","DOI":"10.1214\/aoms\/1177704481","article-title":"A generalization of the gamma distribution","volume":"33","author":"Stacy","year":"1962","journal-title":"Ann. Math. Stat."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1080\/03610929308831010","article-title":"Family of transformed chi-square distributions","volume":"22","author":"Rahman","year":"1992","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1080\/07474946.2014.916928","article-title":"Asymptotic Non-Deficiency of Some Procedures for a Particular Exponential Family Under Relative LINEX Loss","volume":"33","author":"Hwang","year":"2014","journal-title":"Seq. Anal."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Kudryavtsev, A., and Shestakov, O. (2021). Asymptotically normal estimators for the parameters of the gamma-exponential distribution. Mathematics, 9.","DOI":"10.3390\/math9030273"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Reyes, J., Barranco-Chamorro, I., Gallardo, D.I., and G\u00f3mez, H.W. (2018). Generalized Modified Slash Birnbaum\u2014Saunders Distribution. Symmetry, 10.","DOI":"10.3390\/sym10120724"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Z\u00f6rnig, P. (2019). On Generalized Slash Distributions: Representation by Hypergeometric Functions. Stats, 2.","DOI":"10.3390\/stats2030026"},{"key":"ref_17","unstructured":"Prudnikov, A., Brychkov, Y.A., and Marichev, O.I. (1986). Integrals and Series, Vols. 1, 2 and 3, Gordon and Breach Science Publishers."},{"key":"ref_18","unstructured":"Casella, G., and Berger, R.L. (2002). Statistical Inference, Duxbury-Thomson Learning. [2nd ed.]."},{"key":"ref_19","unstructured":"Polovko, A. (1968). Fundamentals of Reliability Theory, Academic Press."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data, John Wiley & Sons. [2nd ed.].","DOI":"10.1002\/9781118033005"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"947","DOI":"10.1109\/TUFFC.2005.1504017","article-title":"Maximum likelihood segmentation of ultrasound images with Rayleigh distribution","volume":"52","author":"Sarti","year":"2005","journal-title":"IEEE Trans. Ultrason. Ferroelectr. Freq. Control"},{"key":"ref_22","first-page":"29","article-title":"Bayesian Reliability Sampling Plans under the Conditions of Rayleigh-Inverse-Rayleigh Distribution","volume":"29","author":"Kalaiselvi","year":"2014","journal-title":"Econ. Qual. Control"},{"key":"ref_23","first-page":"78","article-title":"Approximations to the Non-Isothermal Distributed Activation Energy Model for Biomass Pyrolysis Using the Rayleigh Distribution","volume":"20","author":"Dhaundiyal","year":"2017","journal-title":"Cta Technol. Agric."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1109\/TIP.2003.818017","article-title":"Modeling SAR Images with a Generalization of the Rayleigh Distribution","volume":"13","author":"Kuruoglu","year":"2004","journal-title":"IEEE Trans. Image Process."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2351","DOI":"10.1080\/03610920701215688","article-title":"An Overview of Asymptotic Properties of Estimators in Truncated Distributions","volume":"36","year":"2007","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Lehmann, E. (1999). Elements of Large-Sample Theory, Springer.","DOI":"10.1007\/b98855"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3220","DOI":"10.1080\/03610926.2017.1353621","article-title":"Modified slashed-Rayleigh distribution","volume":"47","author":"Iriarte","year":"2018","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Olmos, N.M., Venegas, O., G\u00f3mez, Y.M., and Iriarte, Y.A. (2019). An Asymmetric Distribution with Heavy Tails and Its Expectation-Maximization (EM) Algorithm Implementation. Symmetry, 11.","DOI":"10.3390\/sym11091150"},{"key":"ref_29","unstructured":"Hogg, R., and Tanis, E. (1993). Probability and Statistical Inference, MacMillan Publishing."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2169","DOI":"10.1080\/STA-200066424","article-title":"Reliability characteristics of the Maxwell distribution: A Bayes estimation study","volume":"34","author":"Bekker","year":"2005","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_31","unstructured":"Johnson, N.L., Kotz, S., and Balakrishnan, N. (1995). Continuous Univariate Distributions, John Wiley & Sons Inc.. [2nd ed.]."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/j.2517-6161.1977.tb01600.x","article-title":"Maximum likelihood from incomplete data via the EM algorithm (with discussion)","volume":"39","author":"Dempster","year":"1977","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_33","unstructured":"R Core Team (2021). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1080\/00401706.1977.10489586","article-title":"The inverse Gaussian distribution as a lifetime model","volume":"19","author":"Chhikara","year":"1977","journal-title":"Technometrics"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/TAC.1974.1100705","article-title":"A new look at the statistical model identification","volume":"19","author":"Akaike","year":"1974","journal-title":"IEE Trans. Autom. Control"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the dimension of a model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Lee, E.T., and Wang, J. (2003). Statistical Methods for Survival Data Analysis, John Wiley & Sons.","DOI":"10.1002\/0471458546"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1226\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:27:57Z","timestamp":1760164077000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/7\/1226"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,8]]},"references-count":37,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["sym13071226"],"URL":"https:\/\/doi.org\/10.3390\/sym13071226","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,8]]}}}