{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T16:47:23Z","timestamp":1771001243435,"version":"3.50.1"},"reference-count":35,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,2,22]],"date-time":"2023-02-22T00:00:00Z","timestamp":1677024000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Umm Al-Qura University","award":["22UQU4310063DSR13"],"award-info":[{"award-number":["22UQU4310063DSR13"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The Shanker distribution, a one-parameter lifetime distribution with an increasing hazard rate function, is recommended by Shanker for modelling lifespan data. In this study, we examine the theoretical and practical implications of 2-component mixture of Shanker model (2-CMSM). A significant feature of proposed model\u2019s hazard rate function is that it has rising, decreasing, and upside-down bathtub forms. We investigate the statistical characteristics of a mixed model, such as the probability-generating function, the factorial-moment-generating function, cumulants, the characteristic function, the Mills ratio, the mean residual life, and the mean time to failure. There is a graphic representation of density, mean, hazard rate functions, coefficient of variation, skewness, and kurtosis. Our final approach is to estimate the parameters of the mixture model using appropriate approaches such as maximum likelihood, least squares, and weighted least squares. Using a simulation analysis, we examined how the estimates behaved graphically. The simulation results demonstrated that, in the majority of cases, the maximum likelihood estimates have the smallest mean square errors among all other estimates. Finally, we observed that when the sample size rises, the precision measures decrease for all of the estimation techniques, indicating that all of the estimation approaches are consistent. Through two real data analyses, the suggested model\u2019s validity and adaptability are contrasted with those of other models, including the mixture of the exponential distributions and the Lindley distributions .<\/jats:p>","DOI":"10.3390\/axioms12030231","type":"journal-article","created":{"date-parts":[[2023,2,23]],"date-time":"2023-02-23T02:01:25Z","timestamp":1677117685000},"page":"231","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Mixture of Shanker Distributions: Estimation, Simulation and Application"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5372-5521","authenticated-orcid":false,"given":"Tahani A.","family":"Abushal","sequence":"first","affiliation":[{"name":"Department of Mathematical Science, Faculty of Applied Science, Umm Al-Qura University, Mecca 24382, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9433-4981","authenticated-orcid":false,"given":"Tabassum Naz","family":"Sindhu","sequence":"additional","affiliation":[{"name":"Department of Statistics, Quaid-i-Azam University, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7149-3314","authenticated-orcid":false,"given":"Showkat Ahmad","family":"Lone","sequence":"additional","affiliation":[{"name":"Department of Basic Science, College of Science and Theoretical Studies, Saudi Electronic University, (Jeddah-M), Riyadh 11673, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marwa K. H.","family":"Hassan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Education, Ain Shams University, Cairo 11566, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7186-7216","authenticated-orcid":false,"given":"Anum","family":"Shafiq","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China"},{"name":"Jiangsu International Joint Laboratory on System Modeling and Data Analysis, Nanjing University of Information Science and Technology, Nanjing 210044, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0167-7152(88)90004-1","article-title":"A finite mixture model for the clustering of mixed-mode data","volume":"6","author":"Everitt","year":"1988","journal-title":"Stat. Probab. Lett."},{"key":"ref_2","unstructured":"Lindsay, B.G. (1995). NSF-CBMS Regional Conference Series in Probability and Statistics, Institute of Mathematical Statistics and the American Statistical Association."},{"key":"ref_3","unstructured":"McLachlan, G.J., and Basford, K.E. (1988). Mixture Models: Inference and Applications to Clustering, M. Dekker."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"McLachlan, G., and Peel, D. (2000). Finite Mixture Models, John Wiley & Sons.","DOI":"10.1002\/0471721182"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1002\/acs.744","article-title":"Bayesian regression and classification using mixtures of Gaussian processes","volume":"17","author":"Shi","year":"2003","journal-title":"Int. J. Adapt. Control. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"251","DOI":"10.12785\/jsap\/030215","article-title":"On the Mixture of BurrXII and Weibull Distribution","volume":"3","author":"Mohammad","year":"2014","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5377","DOI":"10.1016\/j.csda.2006.09.016","article-title":"Mixture of two inverse Weibull distributions: Properties and estimation","volume":"51","author":"Sultan","year":"2007","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/S0951-8320(01)00030-8","article-title":"Models involving two inverse Weibull distributions","volume":"73","author":"Jiang","year":"2001","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"417","DOI":"10.18187\/pjsor.v10i4.620","article-title":"Bayesian prediction under a finite mixture of generalized exponential lifetime model","volume":"10","author":"Mohamed","year":"2014","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1007\/s00180-012-0323-3","article-title":"Using mixture of Gamma distributions for Bayesian analysis in an M\/G\/1 queue with optional second service","volume":"28","author":"Mohammadi","year":"2013","journal-title":"Comput. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1007\/s00362-012-0480-z","article-title":"Maximum likelihood estimation under a finite mixture of generalized exponential distributions based on censored data","volume":"55","author":"Ateya","year":"2014","journal-title":"Stat. Pap."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"17","DOI":"10.18187\/pjsor.v10i1.649","article-title":"Preference of prior for Bayesian analysis of the mixed Burr type X distribution under type I censored samples","volume":"10","author":"Sindhu","year":"2014","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_13","first-page":"1","article-title":"Mixture of Two One-Parameter Lindley Distributions: Properties and Estimation","volume":"15","author":"Daghestani","year":"2021","journal-title":"J. Stat. Theory Pract."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1177\/0142331215578690","article-title":"Bayes estimation of Gumbel mixture models with industrial applications","volume":"38","author":"Sindhu","year":"2016","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_15","first-page":"423","article-title":"A simulation study of parameters for the censored shifted Gompertz mixture distribution: A Bayesian approach","volume":"19","author":"Sindhu","year":"2016","journal-title":"J. Stat. Manag. Syst."},{"key":"ref_16","unstructured":"Sindhu, T.N., Feroze, N., Aslam, M., and Shafiq, A. (2020). Bayesian inference of mixture of two Rayleigh distributions: A new look. Punjab Univ. J. Math., 48."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"587","DOI":"10.4038\/jnsfsr.v46i4.8633","article-title":"Bayesian inference from the mixture of half-normal distributions under censoring","volume":"46","author":"Sindhu","year":"2018","journal-title":"J. Natl. Sci. Found. Sri Lanka"},{"key":"ref_18","first-page":"459","article-title":"Parameter and reliability estimation of inverted Maxwell mixture model","volume":"22","author":"Sindhu","year":"2019","journal-title":"J. Stat. Manag. Syst."},{"key":"ref_19","first-page":"515","article-title":"Mixture of the inverse Rayleigh distribution: Properties and estimation in a Bayesian framework","volume":"39","author":"Ali","year":"2015","journal-title":"Appl. Math."},{"key":"ref_20","first-page":"1","article-title":"Finite mixture models and their applications: A review","volume":"2","author":"Zhang","year":"2015","journal-title":"Austin Biom. Biostat."},{"key":"ref_21","first-page":"338","article-title":"Shanker Distribution and Its Applications","volume":"5","author":"Shanker","year":"2015","journal-title":"Int. J. Stat. Appl."},{"key":"ref_22","first-page":"00121","article-title":"On discrete Poisson-Shanker distribution and its applications","volume":"5","author":"Shanker","year":"2017","journal-title":"Biom. Biostat. Int. J."},{"key":"ref_23","first-page":"1","article-title":"On modeling of lifetime data using one parameter Akash, Lindley and exponential distributions","volume":"3","author":"Shanker","year":"2016","journal-title":"Biom. Biostat. Int. J."},{"key":"ref_24","first-page":"1","article-title":"On modeling of Lifetimes data using exponential and Lindley distributions","volume":"2","author":"Shanker","year":"2015","journal-title":"Biom. Biostat. Int. J."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"8118","DOI":"10.1080\/03610918.2016.1267752","article-title":"Exponentiated Chen distribution: Properties and estimation","volume":"46","author":"Dey","year":"2017","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1080\/01919512.2017.1308817","article-title":"A new extension of generalized exponential distribution with application to ozone data","volume":"39","author":"Dey","year":"2017","journal-title":"Ozone Sci. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1080\/02664763.2016.1268571","article-title":"Poisson exponential distribution: Different methods of estimation","volume":"45","author":"Rodrigues","year":"2018","journal-title":"J. Appl. Stat."},{"key":"ref_28","first-page":"839","article-title":"Statistical properties and different methods of estimation of Gompertz distribution with application","volume":"21","author":"Dey","year":"2018","journal-title":"J. Stat. Manag. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2094","DOI":"10.1007\/s40314-017-0441-1","article-title":"Kumaraswamy distribution: Different methods of estimation","volume":"37","author":"Dey","year":"2017","journal-title":"Comput. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Shafiq, A., Sindhu, T.N., Dey, S., Lone, S.A., and Abushal, T.A. (2023). Statistical Features and Estimation Methods for Half-Logistic Unit-Gompertz Type-I Model. Mathematics, 11.","DOI":"10.3390\/math11041007"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1080\/00949658808811068","article-title":"Least-squares estimation of distribution functions in Johnson\u2019s translation system","volume":"29","author":"Swain","year":"1988","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1080\/00949650108812098","article-title":"Generalized exponential distribution: Different method of estimations","volume":"69","author":"Gupta","year":"2001","journal-title":"J. Stat. Simul."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1016\/j.matcom.2007.06.007","article-title":"Lindley distribution and its application","volume":"78","author":"Ghitany","year":"2008","journal-title":"Math. Comput. Simul."},{"key":"ref_34","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_35","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"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/3\/231\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:39:52Z","timestamp":1760121592000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/3\/231"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,22]]},"references-count":35,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2023,3]]}},"alternative-id":["axioms12030231"],"URL":"https:\/\/doi.org\/10.3390\/axioms12030231","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,22]]}}}