{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T04:30:15Z","timestamp":1787027415379,"version":"build-2736575974"},"reference-count":51,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T00:00:00Z","timestamp":1786579200000},"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>Entropy is a fundamental measure of uncertainty in reliability and lifetime analysis, and the structure of the observed data intrinsically influences its estimation. In many practical applications, inference relies on ordered or record-based samples, for which generalized order statistics and their dual form provide a comprehensive and unifying framework encompassing order statistics, reversed order statistics, and record values as special cases. Despite substantial progress in entropy estimation and lifetime modeling, little attention has been devoted to a unified entropy estimation framework for flexible lifetime models under generalized ordered sampling schemes. This paper investigates the impact of generalized order statistics and their dual form on the estimation of entropy measures for the exponentiated generalized Pham distribution. The proposed distribution extends the classical Pham model through additional shape flexibility, enabling it to accommodate diverse reliability behaviors and heterogeneous tail characteristics. Several fundamental properties are derived, and maximum likelihood estimation and corresponding confidence intervals for model parameters and entropy measures are developed under both generalized order statistics and dual generalized order statistics frameworks. The general results are further specialized to order statistics, reversed order statistics, and upper and lower record values. Applications to two real datasets demonstrate that the proposed distribution provides an excellent fit compared with competing models, as confirmed by goodness-of-fit measures. The findings underscore the pivotal structural role of generalized order statistics and their dual form in entropy-based inference, particularly for record or partially observed data, where the sampling design critically affects uncertainty quantification.<\/jats:p>","DOI":"10.3390\/e28080909","type":"journal-article","created":{"date-parts":[[2026,8,13]],"date-time":"2026-08-13T17:01:35Z","timestamp":1786640495000},"page":"909","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Impact of Generalized Order Statistics and Its Dual on Entropy Estimation for the Exponentiated Generalized Pham Distribution: Theory and Applications"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7040-5623","authenticated-orcid":false,"given":"Zakiah I.","family":"Kalantan","sequence":"first","affiliation":[{"name":"Department of Statistics, Faculty of Sciences, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4512-9866","authenticated-orcid":false,"given":"Sulafah M. S.","family":"Binhimd","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Sciences, King Abdulaziz University, Jeddah 21589, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-7299-0305","authenticated-orcid":false,"given":"Asmaa M.","family":"Abd AL-Fattah","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Commerce, Al-Azhar University (Girls\u2019 Branch), Cairo 11651, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2254-2954","authenticated-orcid":false,"given":"Asmaa A.","family":"Ahmed","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Commerce, Al-Azhar University (Girls\u2019 Branch), Cairo 11651, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5662-5868","authenticated-orcid":false,"given":"Gannat R.","family":"AL-Dayian","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Commerce, Al-Azhar University (Girls\u2019 Branch), Cairo 11651, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3247-2013","authenticated-orcid":false,"given":"Abeer A.","family":"EL-Helbawy","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Commerce, Al-Azhar University (Girls\u2019 Branch), Cairo 11651, Egypt"},{"name":"Department of Basic Sciences, Badr Institute of Science and Technology (BIS), Badr 11829, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0646-9443","authenticated-orcid":false,"given":"Mervat K.","family":"Abd Elaal","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Commerce, Al-Azhar University (Girls\u2019 Branch), Cairo 11651, Egypt"},{"name":"Higher Institute of Marketing, Commerce & Information Systems (MCI), Cairo 11856, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_2","first-page":"547","article-title":"On measures of entropy and information","volume":"Volume 4","year":"1961","journal-title":"Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics"},{"key":"ref_3","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_4","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1214\/11-EJS614","article-title":"Uniform-in-bandwidth consistency for kernel-type estimators of Shannon\u2019s entropy","volume":"5","author":"Bouzebda","year":"2011","journal-title":"Electron. J. Stat."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.1016\/j.spl.2013.01.012","article-title":"Estimation of the Shannon\u2019s entropy of several shifted exponential populations","volume":"83","author":"Kayal","year":"2013","journal-title":"Stat. Probab. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"279","DOI":"10.2991\/jsta.d.210610.001","article-title":"Estimation of entropy for Weibull distribution based on record values","volume":"20","author":"Chacko","year":"2021","journal-title":"J. Stat. Theory Appl."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Al-Babtain, A.A., Elbatal, I., Chesneau, C., and Elgarhy, M. (2021). Estimation of different types of entropies for the Kumaraswamy distribution. PLoS ONE, 16.","DOI":"10.1371\/journal.pone.0249027"},{"key":"ref_8","first-page":"95","article-title":"Entropy Bayesian estimation for Lomax distribution based on record","volume":"19","author":"Hassan","year":"2021","journal-title":"Thail. Stat."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Ren, H., and Hu, X. (2023). Bayesian estimations of Shannon entropy and R\u00e9nyi entropy of inverse Weibull distribution. Mathematics, 11.","DOI":"10.20944\/preprints202305.1037.v1"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"131","DOI":"10.18576\/jsap\/130109","article-title":"Estimation of Kumaraswamy distribution parameters using the principle of maximum entropy","volume":"13","author":"Mahmoud","year":"2024","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_11","first-page":"e70171","article-title":"A Novel entropy-transformed inverse Weibull distribution: Development, properties, and application in diverse data modeling","volume":"7","author":"Sindhu","year":"2025","journal-title":"Eng. Rep."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0378-3758(94)00147-N","article-title":"A concept of generalized order statistics","volume":"48","author":"Kamps","year":"1995","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_13","first-page":"353","article-title":"Recurrence relations for single and product moments of lower generalized order statistics from the inverse Weibull distribution","volume":"34","author":"Pawlas","year":"2001","journal-title":"Demonstr. Math."},{"key":"ref_14","first-page":"13","article-title":"Dual generalized order statistics","volume":"LX1","author":"Burkschat","year":"2003","journal-title":"Int. J. Stat."},{"key":"ref_15","first-page":"125","article-title":"Moments of lower generalized order statistics from power function distribution and its char-815 acterization","volume":"11","author":"Athar","year":"2011","journal-title":"Int. J. Stat. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"189","DOI":"10.12785\/jsap\/030209","article-title":"Recurrence relations for moments of dual generalized order statistics from Weibull Gamma distribution and its characterizations","volume":"3","author":"Mahmoud","year":"2014","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/j.jksus.2015.06.006","article-title":"Dual generalized order statistics from family of J-shaped distribution and its characterization","volume":"27","author":"Khan","year":"2015","journal-title":"J. King Saud Univ.-Sci."},{"key":"ref_18","first-page":"683","article-title":"Statistical inference for Burr Type III distribution on dual generalized order statistics and real data analysis","volume":"10","author":"Kim","year":"2016","journal-title":"Appl. Math. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"402","DOI":"10.2991\/jsta.d.191104.001","article-title":"Relations for moments of dual generalized order statistics from exponentiated Rayleigh distribution and associated inference","volume":"18","author":"Khan","year":"2019","journal-title":"J. Stat. Theory Appl."},{"key":"ref_20","first-page":"649","article-title":"Inference for inverted Kumaraswamy distribution based on dual generalized order statistics","volume":"16","year":"2020","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"531","DOI":"10.18187\/pjsor.v17i3.3338","article-title":"A study on moments of dual generalized order statistics from exponentiated generalized class of distributions","volume":"17","author":"Athar","year":"2021","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"571","DOI":"10.18576\/jsap\/130140","article-title":"Residual and past entropies of concomitants from Lai and Xie extensions of case-II of generalized order statistics and its dual","volume":"13","author":"Gamal","year":"2024","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"7","DOI":"10.12691\/ajams-2-1-2","article-title":"Characterization of distribution by conditional expectation of lower record values","volume":"2","author":"Khan","year":"2014","journal-title":"Am. J. Appl. Math. Stat."},{"key":"ref_24","first-page":"406","article-title":"The role of record values in statistical inference: A review article","volume":"19","author":"Alsanea","year":"2024","journal-title":"Reliab. Theory Appl."},{"key":"ref_25","first-page":"1","article-title":"A vtub-shaped hazard rate function with applications to system safety","volume":"3","author":"Pham","year":"2002","journal-title":"Int. J. Reliab. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Pham, H. (2007). System Software Reliability, Springer Science & Business Media.","DOI":"10.1007\/1-84628-295-0"},{"key":"ref_27","first-page":"79","article-title":"Analysis of Pham (loglog) reliability model using Bayesian approach","volume":"1","author":"Srivastava","year":"2011","journal-title":"Comput. Sci."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1080\/02331934.2013.854787","article-title":"A new software reliability model with Vtub-shaped fault-detection rate and the uncertainty of operating environments","volume":"63","author":"Pham","year":"2014","journal-title":"Optimization"},{"key":"ref_29","first-page":"151","article-title":"A note on the new Pham\u2019s software reliability model","volume":"27","author":"Golev","year":"2019","journal-title":"Neural Parallel Sci. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"889","DOI":"10.1080\/02664763.2021.2001442","article-title":"The unit log\u2013log distribution: A new unit distribution with alternative quantile regression modeling and educational measurements applications","volume":"50","author":"Korkmaz","year":"2023","journal-title":"J. Appl. Stat."},{"key":"ref_31","first-page":"100961","article-title":"A new extended Pham distribution for modelling cancer data","volume":"17","author":"Alotaibi","year":"2024","journal-title":"J. Radiat. Res. Appl. Sci."},{"key":"ref_32","first-page":"e70323","article-title":"Classical and Bayesian methodology for a new inverse statistical model","volume":"7","author":"Elbatal","year":"2025","journal-title":"Eng. Rep."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.6339\/JDS.2013.11(1).1086","article-title":"The exponentiated generalized class of distributions","volume":"11","author":"Cordeiro","year":"2013","journal-title":"J. Data Sci."},{"key":"ref_34","first-page":"47","article-title":"Statistical properties of the exponentiated generalized inverted exponential distribution","volume":"4","author":"Oguntunde","year":"2014","journal-title":"Appl. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"441","DOI":"10.18187\/pjsor.v11i4.1164","article-title":"The transmuted exponentiated generalized-G family of distributions","volume":"11","author":"Yousof","year":"2015","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_36","first-page":"393","article-title":"The exponentiated generalized extended exponential distribution","volume":"14","author":"Bourguignon","year":"2016","journal-title":"J. Data Sci."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"575","DOI":"10.18187\/pjsor.v13i3.2069","article-title":"Inference for exponentiated general class of distributions based on record values","volume":"XIII","author":"Sindi","year":"2017","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1016\/j.jksus.2017.09.009","article-title":"Exponentiated generalized exponential Dagum distribution","volume":"31","author":"Nasiru","year":"2019","journal-title":"J. King Saud Univ.-Sci."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"517","DOI":"10.18187\/pjsor.v16i3.3415","article-title":"Exponentiated exponential-exponentiated Weibull linear mixed distribution: Properties and applications","volume":"16","author":"Abbas","year":"2020","journal-title":"Pak. J. Stat. Oper. Res."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"e04653","DOI":"10.1016\/j.heliyon.2020.e04653","article-title":"The exponentiated generalized power series: Family of distributions: Theory, properties and applications","volume":"6","author":"Oluyede","year":"2020","journal-title":"Heliyon"},{"key":"ref_41","unstructured":"Ali, M.M., Ali, I., Yousof, H.M., and Ibrahim, M. (2023). Exponentiated Generalized General Class of Inverted Distributions Estimation and Prediction. G Families of Probability Distributions: Theory and Practices, Taylor and Francis."},{"key":"ref_42","first-page":"e01719","article-title":"A new extension of the exponentiated generalized-G family of distributions","volume":"20","author":"Keganne","year":"2023","journal-title":"Sci. Afr."},{"key":"ref_43","first-page":"57","article-title":"Exponentiated generalized Weibull exponential distribution: Properties, estimation and applications","volume":"3","author":"Abonongo","year":"2024","journal-title":"Comput. J. Math. Stat. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Binhimd, S.M., Kalantan, Z.I., Abd AL-Fattah, A.M., EL-Helbawy, A.A., AL-Dayian, G.R., Abd EL-Kader, R.E., and Abd Elaal, M.K. (2024). Exponentiated generalized xgamma distribution based on dual generalized order statistics: A Bayesian and maximum likelihood approach. Symmetry, 16.","DOI":"10.3390\/sym16121708"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1090\/S0025-5718-1985-0777276-4","article-title":"The generalized integro-exponential function","volume":"44","author":"Milgram","year":"1985","journal-title":"Math. Comput."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Astorga, J.M., Reyes, J., Santoro, K.I., Venegas, O., and G\u00f3mez, H.W. (2020). A reliability model based on the incomplete generalized integro-exponential function. Mathematics, 8.","DOI":"10.3390\/math8091537"},{"key":"ref_47","unstructured":"ForeCA Package (2025, October 05). Continuous Entropy: Differential Entropy Estimation. CRAN R Project. Available online: https:\/\/search.r-project.org\/CRAN\/refmans\/ForeCA\/html\/continuous_entropy.html."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"113","DOI":"10.12921\/cmst.2007.13.02.113-120","article-title":"Makeham\u2019s generalised distribution","volume":"13","author":"Sylwia","year":"2007","journal-title":"Comput. Methods Sci. Technol."},{"key":"ref_49","unstructured":"World Bank (2025, November 03). World Development Indicators: Poverty Gap at US $3.00 a Day (2021 PPP) (% of Population). Available online: https:\/\/data.worldbank.org\/."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1109\/TR.1987.5222310","article-title":"How to identify a bathtub hazard rate","volume":"36","author":"Aarset","year":"1987","journal-title":"IEEE Trans. Reliab."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"26","DOI":"10.9734\/ajpas\/2023\/v23i4510","article-title":"An extension of the Chen distribution: Properties, simulation study and applications to data","volume":"23","author":"Acquah","year":"2023","journal-title":"Asian J. Probab. Stat."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/909\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T04:12:14Z","timestamp":1787026334000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/8\/909"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,13]]},"references-count":51,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2026,8]]}},"alternative-id":["e28080909"],"URL":"https:\/\/doi.org\/10.3390\/e28080909","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,13]]}}}