{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T10:05:58Z","timestamp":1766570758241,"version":"3.48.0"},"reference-count":57,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T00:00:00Z","timestamp":1766534400000},"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>Herein, we consider the significance of cumulative residual entropy (CRE) and its numerous generalizations. This article presents an extension of the cumulative residual inaccuracy to k-record values. We examine certain properties of this measure. Additionally, we investigate some stochastic ordering and identify the proposed measure for several distributions that frequently arise in various realistic scenarios and have applications across multiple fields of science and engineering.<\/jats:p>","DOI":"10.3390\/e28010017","type":"journal-article","created":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T09:58:34Z","timestamp":1766570314000},"page":"17","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Some Results on Cumulative Residual Inaccuracy Measure of k-Record Values"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5597-3091","authenticated-orcid":false,"given":"Ritu","family":"Goel","sequence":"first","affiliation":[{"name":"School of Engineering and Technology, Vivekananda Institute of Professional Studies-Technical Campus, Pitampura, Delhi 110034, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9572-7952","authenticated-orcid":false,"given":"Vikas","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Applied Sciences, UIET, M.D. University, Rohtak 124001, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-5911-9503","authenticated-orcid":false,"given":"Sarang","family":"Vehale","sequence":"additional","affiliation":[{"name":"School of Cyber Security & Digital Forensics, National Forensic Sciences University, Delhi 110085, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0228-1673","authenticated-orcid":false,"given":"Tony C.","family":"Scott","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physikalische Chemie, RWTH-Aachen University, 52056 Aachen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,24]]},"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","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1111\/j.2517-6161.1961.tb00404.x","article-title":"Inaccuracy and inference","volume":"23","author":"Kerridge","year":"1961","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1220","DOI":"10.1109\/TIT.2004.828057","article-title":"Cumulative residual entropy: A new measure of information","volume":"50","author":"Rao","year":"2004","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1931","DOI":"10.1016\/j.jspi.2006.06.035","article-title":"On the dynamic cumulative residual entropy","volume":"137","author":"Asadi","year":"2007","journal-title":"J. Stat. Plann. Inference"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4072","DOI":"10.1016\/j.jspi.2009.05.038","article-title":"On cumulative entropies","volume":"139","author":"Longobardi","year":"2009","journal-title":"J. Stat. Plann. Inference"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1080\/02331888.2010.494730","article-title":"Dynamic cumulative residual R\u00e9nyi\u2019s entropy","volume":"46","author":"Sunoj","year":"2012","journal-title":"Statistics"},{"key":"ref_7","unstructured":"Taneja, H.C., and Kumar, V. (2012, January 4\u20136). On dynamic cumulative residual inaccuracy measure. Proceedings of the World Congress of Engineering (WCE), London, UK."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"187","DOI":"10.4064\/am-15-2-187-190","article-title":"Limiting properties of the kth record values","volume":"15","author":"Dziubdziela","year":"1976","journal-title":"Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Eskandarzadeh, M., Di Crescenzo, A., and Tahmasebi, S. (2019). Cumulative Measure of Inaccuracy and Mutual Information in k-th Lower Record Values. Mathematics, 7.","DOI":"10.3390\/math7020175"},{"key":"ref_10","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. Plann. Inference"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/0378-3758(94)00062-X","article-title":"k-Record values and the extreme-value index","volume":"45","author":"Berred","year":"1995","journal-title":"J. Stat. Plann. Inference"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"798","DOI":"10.1016\/j.spl.2012.01.004","article-title":"Characterizations of symmetric distributions based on R\u00e9nyi entropy","volume":"82","author":"Fashandi","year":"2012","journal-title":"Stat. Probab. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Arnold, B.C., Balakrishnan, N., and Nagaraja, H.N. (1998). Records, Wiley.","DOI":"10.1002\/9781118150412"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1007\/s00184-012-0408-6","article-title":"Generalized cumulative residual entropy and record values","volume":"76","author":"Psarrakos","year":"2013","journal-title":"Metrika"},{"key":"ref_15","first-page":"107","article-title":"The relevation transform and a generalization of the gamma distribution function","volume":"7","author":"Krakowski","year":"1973","journal-title":"RAIRO-Oper. Res.-Rech. Op\u00e9rationnelle"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1007\/s00184-018-0681-0","article-title":"Reliability properties of proportional hazards relevation transform","volume":"82","author":"Sankaran","year":"2019","journal-title":"Metrika"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1016\/j.spl.2017.02.036","article-title":"Generalized cumulative entropy based on kth lower record values","volume":"126","author":"Tahmasebi","year":"2017","journal-title":"Stat. Probab. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1016\/j.physa.2018.02.200","article-title":"Measure of entropy for past lifetime and record statistics","volume":"503","author":"Goel","year":"2018","journal-title":"Phys. A"},{"key":"ref_19","unstructured":"Weisstein, E.W. (2025, December 06). Hazard Function. From MathWorld\u2014A Worlfram Resource. Available online: https:\/\/mathworld.wolfram.com\/HazardFunction.html."},{"key":"ref_20","unstructured":"Liberto, D. (2025, November 29). Hazard Rate: Definition, How to Calculate, and Example. Available online: https:\/\/www.investopedia.com\/terms\/h\/hazard-rate.asp."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/BF02418571","article-title":"Sur les fonctions convexes et les in\u00e9galit\u00e9s entre les valeurs moyennes","volume":"30","author":"Jensen","year":"1906","journal-title":"Acta Math."},{"key":"ref_22","first-page":"151","article-title":"Measure of inaccuracy and k-record statistics","volume":"110","author":"Goel","year":"2018","journal-title":"Bull. Calcutta Math. Soc."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.7153\/jmi-10-01","article-title":"On the sum of squared logarithms inequality and related inequalities","volume":"10","author":"Dannan","year":"2016","journal-title":"J. Math. Inequal."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Cameron, A.C., and Trivedi, P.K. (2005). Preface. Microeconometrics, Cambridge University Press.","DOI":"10.1017\/CBO9780511811241"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Shaked, M., and Shanthikumar, J.G. (2007). Stochastic Orders, Springer.","DOI":"10.1007\/978-0-387-34675-5"},{"key":"ref_26","unstructured":"Bernardin, L., Chin, P., DeMarco, P., Geddes, K.O., Hare, D.E.G., Heal, K.M., Labahn, G., May, J.P., McCarron, J., and Monagan, M.B. (2012). Maple Programming Guide, Maplesoft, Inc."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Kribs, D.W., Pasieka, A., and \u017byczkowski, K. (2008). Entropy of a Quantum Error Correction Code, World Scientific.","DOI":"10.1142\/S1230161208000225"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1098\/rspa.1998.0160","article-title":"Information-Theoretic Approach to Quantum Error Correction","volume":"454","author":"Nielsen","year":"1998","journal-title":"Proc. R. Soc. A"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.physa.2014.02.070","article-title":"An Entropic Analysis of Approximate Quantum Error Correction","volume":"404","author":"Cafaro","year":"2014","journal-title":"Phys. A"},{"key":"ref_30","unstructured":"Bueno, V.D.C., and Balakrishnan, N. (2020). A Cumulative Residual Inaccuracy Measure for Coherent Systems, Cambridge University Press."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"030505","DOI":"10.1103\/PhysRevLett.125.030505","article-title":"Quantum Error Correction in Scrambling Dynamics and Open Systems","volume":"125","author":"Choi","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"104306","DOI":"10.1103\/PhysRevB.103.104306","article-title":"Statistical Mechanics of Quantum Error Correcting Codes","volume":"103","author":"Li","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1197","DOI":"10.22331\/q-2023-12-05-1197","article-title":"Fitting Quantum Noise Models to Tomography Data","volume":"7","author":"Onorati","year":"2023","journal-title":"Quantum J."},{"key":"ref_34","first-page":"2594","article-title":"Exponentiated Weibull Fading Channel Model in Free-Space Optics","volume":"61","author":"Barrios","year":"2013","journal-title":"IEEE Trans. Commun."},{"key":"ref_35","unstructured":"Preskill, J. (2025, December 21). Quantum Error Correction. Caltech Lecture Notes for Physics 229: Quantum Information and Computation, Chapter 7. Available online: https:\/\/www.preskill.caltech.edu\/ph229\/notes\/chap7.pdf."},{"key":"ref_36","unstructured":"Golse, F., Jin, S., and Liu, N. (2022). Quantum Algorithms for Uncertainty Quantification. arXiv."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Wirsching, G. (2022). Quantum-Inspired Uncertainty Quantification. Front. Comput. Sci., 4.","DOI":"10.3389\/fcomp.2021.662632"},{"key":"ref_38","unstructured":"Engineering and Physical Sciences Research Council (2025). Quantum Uncertainty Quantification (QUQ) Project Overview, University of Exeter."},{"key":"ref_39","unstructured":"Halpern, N., and Hayden, P. (2019). Entropic uncertainty relations in quantum information theory. arXiv."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1038\/nphys1734","article-title":"The uncertainty principle in the presence of quantum memory","volume":"6","author":"Berta","year":"2010","journal-title":"Nat. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"025009","DOI":"10.1088\/1367-2630\/12\/2\/025009","article-title":"Entropic uncertainty relations\u2014A survey","volume":"12","author":"Wehner","year":"2010","journal-title":"New J. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"152","DOI":"10.2307\/2372390","article-title":"A Note on Entropy","volume":"79","author":"Hirschman","year":"1957","journal-title":"Am. J. Math."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"159","DOI":"10.2307\/1970980","article-title":"Inequalities in Fourier Analysis","volume":"102","author":"Beckner","year":"1975","journal-title":"Ann. Math."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/BF01608825","article-title":"Uncertainty relations for information entropy in wave mechanics","volume":"44","author":"Mycielski","year":"1975","journal-title":"Commun. Math. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/0003-4916(76)90057-9","article-title":"Nonlinear wave mechanics","volume":"100","author":"Mycielski","year":"1976","journal-title":"Ann. Phys."},{"key":"ref_46","first-page":"591","article-title":"Generalized Lagrange Multipliers in the Problem of Boltzmann\u2019s Entropy","volume":"66","author":"Everett","year":"1957","journal-title":"Ann. Math."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1103","DOI":"10.1103\/PhysRevLett.60.1103","article-title":"Generalized entropic uncertainty relations","volume":"60","author":"Maassen","year":"1988","journal-title":"Phys. Rev. Lett."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"032101","DOI":"10.1103\/PhysRevA.101.032101","article-title":"Experimental investigation of entropic uncertainty relations and coherence uncertainty relations","volume":"101","author":"Ding","year":"2020","journal-title":"Phys. Rev. A"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"261","DOI":"10.5506\/APhysPolB.42.261","article-title":"Spontaneous symmetry breaking and mass generation as built-in phenomena in logarithmic nonlinear quantum theory","volume":"42","author":"Zloshchastiev","year":"2011","journal-title":"Acta Phys. Pol. B"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"2530008","DOI":"10.1142\/S0217979225300087","article-title":"Origins of logarithmic nonlinearity in the theory of fluids and beyond","volume":"39","author":"Zloshchastiev","year":"2025","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Scott, T.C., and Glasser, M.L. (2025). Kink Soliton Solutions in the Logarithmic Schr\u00f6dinger Equation. Mathematics, 13.","DOI":"10.3390\/math13050827"},{"key":"ref_52","unstructured":"(2025, December 21). Athletics Australia. Australian Para Athletics Records, 2025. Available online: https:\/\/www.athletics.com.au\/results-records-toplists\/records\/."},{"key":"ref_53","unstructured":"(2025, December 21). USA Track & Field. Record Applications. Available online: https:\/\/usatf.org\/resources\/statistics\/record-applications."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Teegavarapu, R. (2019). Statistical Analysis of Hydrologic Variables, Tufts University. Technical report.","DOI":"10.1061\/9780784415177"},{"key":"ref_55","first-page":"15","article-title":"The value of streamflow record augmentation procedures for estimating low-flow statistics","volume":"127","author":"Vogel","year":"1991","journal-title":"J. Hydrol."},{"key":"ref_56","unstructured":"Dey, R. (2015). Weather forecasting using Convex Hull & K-Means clustering. arXiv."},{"key":"ref_57","first-page":"5574","article-title":"Dispersion indices based on Kerridge inaccuracy measure and applications","volume":"53","author":"Balakrishnan","year":"2023","journal-title":"Comput. Stat."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/17\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,24]],"date-time":"2025-12-24T10:02:42Z","timestamp":1766570562000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,24]]},"references-count":57,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["e28010017"],"URL":"https:\/\/doi.org\/10.3390\/e28010017","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,24]]}}}