{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T04:22:56Z","timestamp":1772252576482,"version":"3.50.1"},"reference-count":59,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,3]],"date-time":"2018-01-03T00:00:00Z","timestamp":1514937600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The asymmetric bimodal exponential power (ABEP) distribution is an extension of the generalized gamma distribution to the real line via adding two parameters that fit the shape of peakedness in bimodality on the real line. The special values of peakedness parameters of the distribution are a combination of half Laplace and half normal distributions on the real line. The distribution has two parameters fitting the height of bimodality, so capacity of bimodality is enhanced by using these parameters. Adding a skewness parameter is considered to model asymmetry in data. The location-scale form of this distribution is proposed. The Fisher information matrix of these parameters in ABEP is obtained explicitly. Properties of ABEP are examined. Real data examples are given to illustrate the modelling capacity of ABEP. The replicated artificial data from maximum likelihood estimates of parameters of ABEP and other distributions having an algorithm for artificial data generation procedure are provided to test the similarity with real data. A brief simulation study is presented.<\/jats:p>","DOI":"10.3390\/e20010023","type":"journal-article","created":{"date-parts":[[2018,1,3]],"date-time":"2018-01-03T12:00:06Z","timestamp":1514980806000},"page":"23","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Asymmetric Bimodal Exponential Power Distribution on the Real Line"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2933-857X","authenticated-orcid":false,"given":"Mehmet","family":"\u00c7ankaya","sequence":"first","affiliation":[{"name":"Department of International Trading, School of Applied Sciences, U\u015fak University, U\u015fak 64200, Turkey"},{"name":"Department of Statistics, Faculty of Arts and Sciences, U\u015fak University, U\u015fak 64200, Turkey"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1080\/02331888.2014.918618","article-title":"A doubly skewed normal distribution","volume":"49","author":"Arnold","year":"2015","journal-title":"Statistics"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1080\/03610926.2013.765475","article-title":"Bimodal symmetric-asymmetric power-normal families","volume":"47","author":"Bolfarine","year":"2013","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_3","first-page":"338","article-title":"The alpha\u2013beta skew normal distribution: Properties and applications","volume":"50","author":"Shafiei","year":"2016","journal-title":"Statistics"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"6098","DOI":"10.1080\/03610926.2015.1117639","article-title":"A new generalization of alpha-skew-normal distribution","volume":"46","author":"Sharafi","year":"2017","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"195","DOI":"10.12785\/jsap\/020302","article-title":"Analyzing skewed data with the epsilon skew gamma distribution","volume":"2","author":"Abdulah","year":"2013","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18576\/jsap\/030101","article-title":"Bimodal class based on the inverted symmetrized gamma distribution with Applications","volume":"3","author":"Abdulah","year":"2014","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1147","DOI":"10.1080\/03610920701713286","article-title":"Gamma mixture: bimodality, inflexions and L-moments","volume":"8","author":"Ahmed","year":"2008","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1080\/03610926.2012.746986","article-title":"Some skew-symmetric distributions which include the bimodal ones","volume":"44","author":"Dexter","year":"2015","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"912","DOI":"10.1080\/03610920902807903","article-title":"An extension of the epsilon-skew-normal distribution","volume":"39","year":"2010","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3838","DOI":"10.1080\/03610926.2011.625488","article-title":"Exponentiated sinh Cauchy distribution with applications","volume":"42","author":"Cooray","year":"2013","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3205","DOI":"10.1080\/00949655.2016.1157180","article-title":"Fitting asymmetric bimodal data with selected distributions","volume":"86","author":"Andrade","year":"2016","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_12","unstructured":"Rathie, P.N., and Swamee, P.K. (2006). On a New Invertible Generalized Logistic Distribution Approximation to Normal Distribution, Department of Statistics, University of Brasilia. Technical Research Report No. 07\/2006."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Rathie, P.N., Silva, P., and Olinto, G. (2016). Applications of skew models using generalized logistic distribution. Axioms, 5.","DOI":"10.3390\/axioms5020010"},{"key":"ref_14","first-page":"224","article-title":"Alpha-skew-normal distribution","volume":"29","year":"2010","journal-title":"Proyecciones (Antofagasta)"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1080\/00949655.2017.1381698","article-title":"Bimodality based on the generalized skew-normal distribution","volume":"88","author":"Venegas","year":"2018","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_16","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_17","doi-asserted-by":"crossref","first-page":"85","DOI":"10.22237\/jmasm\/1083370200","article-title":"Beta-normal distribution: Bimodality properties and application","volume":"3","author":"Famoye","year":"2004","journal-title":"J. Mod. Appl. Stat. Methods"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10260-014-0258-7","article-title":"A generalization of the slashed distribution via alpha skew normal distribution","volume":"23","author":"Gui","year":"2014","journal-title":"Stat. Methods Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1002\/env.1026","article-title":"Bimodal extension based on the skew-normal distribution with application to pollen","volume":"22","author":"Salinas","year":"2011","journal-title":"Environmetrics"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/s00362-009-0240-x","article-title":"Generalized skew two-piece skew-normal distribution","volume":"52","author":"Jamalizadeh","year":"2011","journal-title":"Stat. Pap."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3722","DOI":"10.1080\/03610926.2011.568156","article-title":"On some properties of the beta normal distribution","volume":"41","author":"Cintra","year":"2012","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.1080\/02331888.2012.688207","article-title":"Inference in flexible families of distributions with normal kernel","volume":"47","author":"Rocha","year":"2013","journal-title":"Statistics"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1080\/03610920701541174","article-title":"The generalized Birnbaum\u2013Saunders distribution and its theory, methodology, and application","volume":"37","author":"Sanhueza","year":"2008","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"774","DOI":"10.1016\/j.spl.2012.11.024","article-title":"Alpha-Skew-Laplace distribution","volume":"83","author":"Shams","year":"2013","journal-title":"Stat. Probab. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"973","DOI":"10.1016\/j.spl.2011.01.017","article-title":"The beta Laplace distribution","volume":"81","author":"Cordeiro","year":"2011","journal-title":"Stat. Probab. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.1080\/03610926.2014.882950","article-title":"Bimodal skew-symmetric normal distribution","volume":"45","author":"Hassan","year":"2016","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1002\/wics.1255","article-title":"Methods for generating families of univariate continuous distributions in the recent decades","volume":"5","author":"Lee","year":"2013","journal-title":"Comput. Stat."},{"key":"ref_28","first-page":"379","article-title":"A Bimodal exponential power distribution","volume":"26","author":"Hassan","year":"2010","journal-title":"Pak. J. Stat."},{"key":"ref_29","first-page":"371","article-title":"A bimodal extension of the generalized gamma distribution","volume":"38","author":"Bulut","year":"2015","journal-title":"Rev. Colomb. Estad."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1016\/j.jspi.2003.11.014","article-title":"Statistical inference for a general class of asymmetric distributions","volume":"128","author":"Gomez","year":"2005","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Singh, V. (1998). Entropy-Based Parameter Estimation in Hydrology, Springer Science+Business Media. Water Science and Technology Library.","DOI":"10.1007\/978-94-017-1431-0"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physa.2017.02.009","article-title":"On statistical properties of Jizba\u2013Arimitsu hybrid entropy","volume":"475","author":"Korbel","year":"2017","journal-title":"Physica A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/S0378-3758(99)00096-8","article-title":"The epsilon-skew-normal distribution for analyzing near-normal data","volume":"83","author":"Mudholkar","year":"2000","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_34","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (2014). Tables of Integrals, Sums, Series, and Products, Nauka."},{"key":"ref_35","unstructured":"Lehmann, E.L., and Casella, G. (1998). Theory of Point Estimation, Wadsworth & Brooks\/Cole."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Embrechts, P., Kluppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance, Springer.","DOI":"10.1007\/978-3-642-33483-2"},{"key":"ref_37","unstructured":"Elsalloukh, H. (2008, January 3\u20137). The Epsilon-Skew Laplace Distribution. Proceedings of the JSM American Statistical Association Conference, Denver, CO, USA."},{"key":"ref_38","first-page":"201","article-title":"Further Results on the Epsilon-Skew Exponential Power Distribution Family","volume":"28","author":"Elsalloukh","year":"2009","journal-title":"Far East J. Theor. Stat."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1080\/02664760500079464","article-title":"A generalized normal distribution","volume":"32","author":"Nadarajah","year":"2005","journal-title":"J. Appl. Stat."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.spl.2003.01.001","article-title":"The extended exponential power distribution and Bayesian robustness","volume":"65","author":"Choy","year":"2003","journal-title":"Stat. Probab. Lett."},{"key":"ref_41","first-page":"359","article-title":"On Bayesian modeling of fat tails and skewness","volume":"93","author":"Steel","year":"1998","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.jeconom.2008.09.038","article-title":"Properties and estimation of asymmetric exponential power distribution","volume":"148","author":"Zhu","year":"2009","journal-title":"J. Econom."},{"key":"ref_43","first-page":"1331","article-title":"Modeling and inference with \u03bd-spherical distributions","volume":"90","author":"Fernandez","year":"1995","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1002\/jae.961","article-title":"Asymmetric power distribution: Theory and applications to risk measurement","volume":"22","author":"Komunjer","year":"2007","journal-title":"J. Appl. Econom."},{"key":"ref_45","unstructured":"Theodossiou, P. (2018, January 02). Skewed Generalized Error Distribution of Financial Assets and Option Pricing. Available online: https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstractid=219679."},{"key":"ref_46","first-page":"97","article-title":"The epsilon-skew exponential power distribution family","volume":"17","author":"Elsalloukh","year":"2005","journal-title":"Far East J. Theor. Stat."},{"key":"ref_47","first-page":"296","article-title":"On the law of frequency of errors","volume":"31","author":"Subbotin","year":"1923","journal-title":"Mat. Sb."},{"key":"ref_48","unstructured":"Box, G.E.P., and Tiao, G.C. (1973). Bayesian Inference in Statistical Analysis, Addison-Wesley."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1","DOI":"10.18637\/jss.v012.i04","article-title":"A Software Tool for the Exponential Power Distribution: The normalp package","volume":"12","author":"Mineo","year":"2005","journal-title":"J. Stat. Softw."},{"key":"ref_50","unstructured":"Johnson, N.L., Kotz, S., and Balakrishnan, N. (1994). Continuous Univariate Distributions, John Wiley & Sons. [2nd ed.]."},{"key":"ref_51","unstructured":"Johnson, N.L., Kotz, S., and Balakrishnan, N. (1995). Continuous Univariate Distributions, John Wiley & Sons. [2nd ed.]."},{"key":"ref_52","doi-asserted-by":"crossref","unstructured":"Mathai, A.M., and Haubold, H.J. (2008). Special Functions for Applied Scientists, Springer.","DOI":"10.1007\/978-0-387-75894-7"},{"key":"ref_53","unstructured":"Horn, R.A., and Johnson, C.R. (2013). Matrix Analysis, Cambridge University Press. [2nd ed.]."},{"key":"ref_54","unstructured":"Prommier, Y., Reinhold, W., Sunshine, M., and Varma, S. (2018, January 02). Genomics and Bioinformatics Group, Available online: http:\/\/discover.nci.nih.gov."},{"key":"ref_55","unstructured":"Tiku, M.L., and Akkaya, A.D. (2004). Robust Estimation and Hypothesis Testing, New Age International."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Conn, A.R., Scheinberg, K., and Vicente, L.N. (2008). Introduction to Derivative-Free Optimization, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9780898718768"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1080\/02331880802401241","article-title":"The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation","volume":"43","author":"Arslan","year":"2009","journal-title":"Statistics"},{"key":"ref_58","first-page":"353","article-title":"Alpha-skew generalized t distribution","volume":"38","author":"Arslan","year":"2015","journal-title":"Rev. Colomb. Estad."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/BF02294361","article-title":"Model selection and Akaike\u2019s Information Criterion (AIC): The general theory and its analytical extensions","volume":"52","author":"Bozdogan","year":"1987","journal-title":"Psychometrika"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/23\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:50:01Z","timestamp":1760194201000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/23"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,3]]},"references-count":59,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010023"],"URL":"https:\/\/doi.org\/10.3390\/e20010023","relation":{"has-preprint":[{"id-type":"doi","id":"10.20944\/preprints201712.0080.v1","asserted-by":"object"}]},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1,3]]}}}