{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T18:32:29Z","timestamp":1775932349624,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,6,30]],"date-time":"2025-06-30T00:00:00Z","timestamp":1751241600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Competing risk models are essential in survival analysis for studying systems with multiple mutually exclusive failure events. This study investigates the application of competing risk models in the presence of progressively Type-II censored data for the Dagum distribution, a flexible distribution suited for modeling data with heavy tails and varying skewness and kurtosis. The methodology includes maximum likelihood estimation of the unknown parameters, with a focus on the special case of a common shape parameter, which allows for a closed-form expression of the relative risks. A hypothesis test is developed to assess the validity of this assumption, and both asymptotic and bootstrap confidence intervals are constructed. The performance of the proposed methods is evaluated through Monte Carlo simulations, and their applicability is demonstrated with a real-world example.<\/jats:p>","DOI":"10.3390\/axioms14070508","type":"journal-article","created":{"date-parts":[[2025,6,30]],"date-time":"2025-06-30T13:06:17Z","timestamp":1751288777000},"page":"508","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Analyzing Competing Risks with Progressively Type-II Censored Data in Dagum Distributions"],"prefix":"10.3390","volume":"14","author":[{"given":"Raghd","family":"Badwan","sequence":"first","affiliation":[{"name":"Statistics Program, Department of Mathematics and Statistics, College of Arts and Sciences, Qatar University, Doha 2713, Qatar"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9777-105X","authenticated-orcid":false,"given":"Reza","family":"Pakyari","sequence":"additional","affiliation":[{"name":"Statistics Program, Department of Mathematics and Statistics, College of Arts and Sciences, Qatar University, Doha 2713, Qatar"}]}],"member":"1968","published-online":{"date-parts":[[2025,6,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/s10985-012-9230-8","article-title":"Applying competing risks regression models: An overview","volume":"19","author":"Haller","year":"2013","journal-title":"Lifetime Data Anal."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1093\/aje\/kwp107","article-title":"Competing Risk Regression Models for Epidemiologic Data","volume":"170","author":"Lau","year":"2009","journal-title":"Am. J. Epidemiol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/j.csda.2020.107125","article-title":"Decrement rates and a numerical method under competing risks","volume":"156","author":"Lee","year":"2021","journal-title":"Comp. Stat. Data Anal."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Ma, Z., and Krings, A.W. (2008, January 1\u20138). Competing Risks Analysis of Reliability, Survivability, and Prognostics and Health Management (PHM). Proceedings of the IEEE Aerospace Conference, Big Sky, MT, USA.","DOI":"10.1109\/AERO.2008.4526636"},{"key":"ref_5","unstructured":"Herd, R.G. (1956). Estimation of Parameters of a Population from a Multi-Censored Sample. [Ph.D. Thesis, Iowa State College]."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1080\/00401706.1963.10490102","article-title":"Progressively censored samples in life testing","volume":"5","author":"Cohen","year":"1963","journal-title":"Technometrics"},{"key":"ref_7","unstructured":"Balakrishnan, N., and Aggrawala, R. (2000). Progressive Censoring: Theory, Methods, and Applications, Birkh\u00e4user."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1007\/s11749-007-0061-y","article-title":"Progressive censoring methodology: An appraisal (with discusssions)","volume":"16","author":"Balakrishnan","year":"2007","journal-title":"TEST"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Balakrishnan, N., and Cramer, E. (2014). The Art of Progressive Censoring, Birkh\u00e4user. Statistics for Industry and Technology.","DOI":"10.1007\/978-0-8176-4807-7"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1016\/S0169-7161(03)23018-2","article-title":"Analysis of progressively censored competing risks data","volume":"23","author":"Kundu","year":"2003","journal-title":"Handb. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"4083","DOI":"10.1016\/j.csda.2009.04.010","article-title":"On progressively censored competing risks data for Weibull distributions","volume":"53","author":"Pareek","year":"2009","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1007\/s13571-011-0024-x","article-title":"Bayesian analysis of progressively censored competing risks data","volume":"73","author":"Kundu","year":"2011","journal-title":"Sankhya B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1285","DOI":"10.1016\/j.csda.2010.09.017","article-title":"Progressively type-II censored competing risks data from Lomax distributions","volume":"55","author":"Cramer","year":"2011","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1016\/j.cam.2018.05.013","article-title":"Inference of progressively censored competing risks data from Kumaraswamy distributions","volume":"343","author":"Wang","year":"2018","journal-title":"J. Comput. Appl. Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cam.2020.112922","article-title":"Statistical inference of Burr-XII distribution under progressive Type-II censored competing risks data with binomial removals","volume":"378","author":"Qin","year":"2020","journal-title":"J. Comput. Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1016\/j.apm.2021.05.008","article-title":"Statistical analysis of adaptive type-II progressively censored competing risks from Weibull models","volume":"98","author":"Ren","year":"2021","journal-title":"Appl. Math. Model."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1080\/03610918.2011.633195","article-title":"Statistical Analysis of the Log-Normal Distribution under Type-II Progressive Hybrid Censoring Schemes","volume":"42","author":"Hemmati","year":"2013","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2506","DOI":"10.1080\/00949655.2013.851681","article-title":"Exact inference for competing risks model with generalized Type I hybrid censored exponential data","volume":"84","author":"Mao","year":"2014","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"108","DOI":"10.14419\/ijasp.v2i2.3423","article-title":"Analysis of Generalized Exponential Distribution Under Adaptive Type-II Progressive Hybrid Censored Competing Risks Data","volume":"2","author":"Ashour","year":"2014","journal-title":"Int. J. Adv. Stat. Prob."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3629","DOI":"10.1080\/00949655.2016.1178747","article-title":"Analysis of progressively Type-I interval censored competing risks data for a class of an exponential distribution","volume":"86","author":"Ahmadi","year":"2016","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"566","DOI":"10.1214\/23-BJPS582","article-title":"Inference for a competing risks model with Burr XII distributions under generalized progressive hybrid censoring","volume":"37","author":"Chandra","year":"2023","journal-title":"Braz. J. Probab. Stat."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/s00180-018-0847-2","article-title":"Bayesian analysis of Weibull distribution based on progressive type-II censored competing risks data with binomial removals","volume":"34","author":"Chacko","year":"2019","journal-title":"Comput. Stat."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1016\/j.cam.2016.01.016","article-title":"Bayes estimation and expected termination time for the competing risks model from Gompertz distribution under progressively hybrid censoring with binomial removals","volume":"300","author":"Wu","year":"2016","journal-title":"J. Comput. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1007\/s00180-017-0725-3","article-title":"Classical and Bayesian Inference in Two Parameter Exponential Distribution with Randomly Censored Data","volume":"33","author":"Krishna","year":"2018","journal-title":"Comput. Stat."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Abd El-Raheem, A.M., Hosny, M., and Abu-Mousa, M.H. (2021). On Progressive Censored Competing Risks Data: Real Data Application and Simulation study. Mathematics, 9.","DOI":"10.3390\/math9151805"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2479","DOI":"10.1134\/S1995080222120149","article-title":"Analysis of Progressive Type-II Competing Risks Data, with Applications","volume":"43","author":"Hassan","year":"2022","journal-title":"Lobachevskii J. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"3867","DOI":"10.1002\/qre.3180","article-title":"Bayesian and non-Bayesian inference of Weibull lifetimemodel based on partially observed competing risks dataunder unified hybrid censoring scheme","volume":"38","author":"Dutta","year":"2022","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1080\/00949655.2023.2249165","article-title":"Parametric inferences using dependent competing risks data with partially observed failure causes from MOBK distribution under unified hybrid censoring","volume":"94","author":"Dutta","year":"2023","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2988","DOI":"10.1080\/03610918.2022.2093373","article-title":"Statistical inference of dependent competing risks from Marshall\u2013Olkin bivariate Burr-XII distribution under complex censoring","volume":"53","author":"Tian","year":"2022","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_30","first-page":"765","article-title":"Inference of a competing risks model with partially observed failure causes under improved adaptive type-II progressive censoring","volume":"37","author":"Dutta","year":"2023","journal-title":"Proc. Inst. Mech. Eng. Part O J. Risk Reliab."},{"key":"ref_31","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_32","first-page":"413","article-title":"A new model of personal income distribution: Specification and estimation","volume":"30","author":"Dagum","year":"1977","journal-title":"Econ. Appl."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Kleiber, C., and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.","DOI":"10.1002\/0471457175"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.15672\/hujms.1482377","article-title":"On some mixtures of the Kies distribution","volume":"53","author":"Zaevski","year":"2024","journal-title":"Hacettepe J. Math. Stat."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"287","DOI":"10.15559\/23-VMSTA227","article-title":"On some composite Kies families: Distributional properties and saturation in Hausdorff sense","volume":"10","author":"Zaevski","year":"2023","journal-title":"Mod. Stochastics Theory Appl."},{"key":"ref_36","first-page":"245","article-title":"A review of more than one hundred Pareto-tail index estimators","volume":"80","author":"Fedotenkov","year":"2020","journal-title":"Statistica"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Cheng, T., Peng, X., Choiruddin, A., He, X., and Chen, K. (2025). Environmental extreme risk modeling via sub-sampling block maxima. arXiv.","DOI":"10.21203\/rs.3.rs-4246942\/v1"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1643","DOI":"10.1081\/STA-200063183","article-title":"Reliability of a stress-strength model with Burr type III distributions","volume":"34","author":"Mokhlis","year":"2005","journal-title":"Commun. Stat Theory Methods"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1007\/s00362-012-0463-0","article-title":"A copula-based approach to account for dependence in stress-strength models","volume":"54","author":"Domma","year":"2013","journal-title":"Stat. Pap."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1093\/imanum\/8.1.141","article-title":"Two-Point Step Size Gradient Methods","volume":"8","author":"Barzilai","year":"1988","journal-title":"IMA J. Num. Anal."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1080\/00031305.1995.10476150","article-title":"A simple simulational algorithm for generating progressive Type-II censored samples","volume":"49","author":"Balakrishnan","year":"1995","journal-title":"Amer. Statist."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/cc6852","article-title":"Risk factors for the development of nosocomial pneumonia and mortality on intensive care units: Application of competing risks models","volume":"12","author":"Wolkewitz","year":"2008","journal-title":"Crit. Care"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1388","DOI":"10.1038\/bmt.2009.359","article-title":"Regression modeling of competing risk using R: An in depth guide for clinicians","volume":"45","author":"Scrucc","year":"2010","journal-title":"Bone Marrow Transp."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.1007\/s00180-018-0804-0","article-title":"Likelihood-based inference for bivariate latent failure time models with competing risks under the generalized FGM copula","volume":"33","author":"Shih","year":"2018","journal-title":"Comput. Stat."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/508\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:01:46Z","timestamp":1760032906000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/508"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,30]]},"references-count":44,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["axioms14070508"],"URL":"https:\/\/doi.org\/10.3390\/axioms14070508","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,30]]}}}