{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T13:03:57Z","timestamp":1765458237926,"version":"3.46.0"},"reference-count":33,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T00:00:00Z","timestamp":1765411200000},"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 present study investigates the problem of determining sharp bounds for key reliability and distributional characteristics associated with order statistics. We establish pointwise sharp two-sided bounds for linear combinations of joint distribution functions and joint reliability functions of selected order statistics based on k-independent and identically distributed random variables. The proposed framework is general and also applies to arbitrarily dependent observations. The obtained results provide exact bounds for the expected values of functions of order statistics corresponding to finite-valued random variables. Furthermore, the study yields the best possible upper and lower bounds for the joint reliability function of semicoherent systems with shared exchangeable k-independent components.<\/jats:p>","DOI":"10.3390\/e27121250","type":"journal-article","created":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T12:57:41Z","timestamp":1765457861000},"page":"1250","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Tight Bounds for Joint Distribution Functions of Order Statistics Under k-Independence"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4726-9499","authenticated-orcid":false,"given":"Andrzej","family":"Okolewski","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Lodz University of Technology, 93-590 Lodz, Poland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5282-8369","authenticated-orcid":false,"given":"Barbara","family":"Blazejczyk-Okolewska","sequence":"additional","affiliation":[{"name":"Division of Dynamics, Lodz University of Technology, 90-537 Lodz, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,11]]},"reference":[{"key":"ref_1","first-page":"497","article-title":"Order Statistics: Applications","volume":"17","author":"Balakrishnan","year":"1998","journal-title":"Handbook of Statistics"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"David, H.A., and Nagaraja, H.N. (2003). Order Statistics, Wiley. [3rd ed.].","DOI":"10.1002\/0471722162"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Rychlik, T. (2001). Projecting Statistical Functionals, Springer.","DOI":"10.1007\/978-1-4612-2094-7"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1190","DOI":"10.1214\/aos\/1176343650","article-title":"The multivariate inclusion\u2013exclusion formula and order statistics from dependent data","volume":"4","author":"Maurer","year":"1976","journal-title":"Ann. Stat."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1214\/aos\/1176344567","article-title":"Bounds on expectations of linear systematic statistics based on dependent samples","volume":"7","author":"Arnold","year":"1979","journal-title":"Ann. Stat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/S0378-3758(00)00167-1","article-title":"Expectation bounds on linear estimators from dependent samples","volume":"93","author":"Papadatos","year":"2001","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1899","DOI":"10.1080\/03610920500200550","article-title":"Bounds for L-statistics from weakly dependent samples of random length","volume":"34","author":"Kaluszka","year":"2005","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.jspi.2003.08.015","article-title":"Sharp bounds for L-statistics from dependent samples of random length","volume":"127","author":"Kaluszka","year":"2005","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"667","DOI":"10.1017\/S0269964806060414","article-title":"Tight bounds on expected order statistics","volume":"20","author":"Bertsimas","year":"2006","journal-title":"Probab. Eng. Inf. Sci."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2421","DOI":"10.1016\/j.jspi.2012.02.019","article-title":"Bounds on dispersion of order statistics based on dependent symmetrically distributed random variables","volume":"142","author":"Rychlik","year":"2012","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"596","DOI":"10.1080\/02331888.2015.1074234","article-title":"Maximizing the expected range from dependent observations under mean\u2013variance information","volume":"50","author":"Papadatos","year":"2016","journal-title":"Statistics"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1137\/1011075","article-title":"Extrema of expectations of uniform order statistics","volume":"11","author":"Mallows","year":"1969","journal-title":"SIAM Rev."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/BF00536046","article-title":"A class of dependent random variables and their maxima","volume":"48","author":"Lai","year":"1978","journal-title":"Z. Wahrsch. Verw. Gebiete"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/0167-7152(92)90105-E","article-title":"Statistically extremal distributions for dependent samples","volume":"13","author":"Rychlik","year":"1992","journal-title":"Stat. Probab. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/0167-7152(92)90071-C","article-title":"Bounds on distribution functions of order statistics for dependent variates","volume":"14","author":"Caraux","year":"1992","journal-title":"Stat. Probab. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/02331888308802385","article-title":"Bounds for expectations of L-estimates for dependent samples","volume":"24","author":"Rychlik","year":"1993","journal-title":"Statistics"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/S0169-7161(98)16008-X","article-title":"Bounds for expectations of L-estimates. Order Statistics: Theory and Methods","volume":"16","author":"Balakrishnan","year":"1998","journal-title":"Handbook of Statistics"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/S0167-7152(01)00014-1","article-title":"Distribution and expectation bounds on order statistics from possibly dependent variates","volume":"4","author":"Papadatos","year":"2001","journal-title":"Stat. Probab. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1080\/02331888.2015.1105804","article-title":"Bounds on expectations of L-estimates for maximally and minimally stable samples","volume":"50","author":"Okolewski","year":"2016","journal-title":"Statistics"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Bene\u0161, V., and \u0160t\u011bp\u00e1n, J. (1997). Bounding moments of an order statistic when each k-tuple is independent. Distributions with Given Marginals and Moment Problems, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-011-5532-8"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1080\/02331888.2017.1289533","article-title":"Distribution bounds for order statistics when each k-tuple has the same piecewise uniform copula","volume":"51","author":"Okolewski","year":"2017","journal-title":"Statistics"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/S0196-8858(02)00562-6","article-title":"Minimizing the expected minimum","volume":"31","author":"Mallows","year":"2003","journal-title":"Adv. Appl. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmva.2017.05.007","article-title":"Extremal properties of order statistic distributions for dependent samples with partially known multidimensional marginals","volume":"160","author":"Okolewski","year":"2017","journal-title":"J. Multivar. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Cram\u00e9r, H. (1946). Mathematical Methods of Statistics, Princeton University Press.","DOI":"10.1515\/9781400883868"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"24","DOI":"10.4153\/CMB-1990-004-6","article-title":"Dependent random variables with independent subsets-II","volume":"33","author":"Wang","year":"1990","journal-title":"Canad. Math. Bull."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1137\/S089548019223872X","article-title":"Chemoff-Hoeffding bounds for applications with limited independence","volume":"8","author":"Schmidt","year":"1995","journal-title":"SIAM J. Disc. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"836","DOI":"10.2307\/3213951","article-title":"A note on the background of several Bonferroni\u2013Galambos-type inequalities","volume":"22","year":"1985","journal-title":"J. Appl. Probab."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1017\/S0515036100006930","article-title":"Rating of largest claims and ECOMOR reinsurance treaties for large portfolios","volume":"13","author":"Kremer","year":"1982","journal-title":"Astin Bulletin"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"559","DOI":"10.1016\/j.ejor.2017.06.022","article-title":"Joint signature of two or more systems with applications to multistate systems made up of two-state components","volume":"263","author":"Marichal","year":"2017","journal-title":"Eur. J. Oper. Res."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1016\/j.jmva.2005.09.003","article-title":"Reliability and expectation bounds for coherent systems with exchangeable components","volume":"98","author":"Navarro","year":"2007","journal-title":"J. Multivar. Anal."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Preparata, F.P., and Shamos, M.I. (1985). Computational Geometry, Springer.","DOI":"10.1007\/978-1-4612-1098-6"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Matou\u0161ek, J. (2002). Lectures on Discrete Geometry, Springer.","DOI":"10.1007\/978-1-4613-0039-7"},{"key":"ref_33","unstructured":"Goodman, J.E., and O\u2019Rourke, J. (2017). Convex hull computations. Handbook of Discrete and Computational Geometry, CRC Press. [3rd ed.]."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/12\/1250\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T12:59:12Z","timestamp":1765457952000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/12\/1250"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,11]]},"references-count":33,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["e27121250"],"URL":"https:\/\/doi.org\/10.3390\/e27121250","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,11]]}}}