Marco Tschimpke , Manuela Schreyer , Wolfgang Trutschnig
{"title":"重新审视斯皮尔曼 ρ 和斯皮尔曼脚规 ϕ 所确定的区域","authors":"Marco Tschimpke , Manuela Schreyer , Wolfgang Trutschnig","doi":"10.1016/j.cam.2024.116259","DOIUrl":null,"url":null,"abstract":"<div><p>Kokol and Stopar (2023) recently studied the exact region <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>ϕ</mi><mo>,</mo><mi>ρ</mi></mrow></msub></math></span> determined by Spearman’s footrule <span><math><mi>ϕ</mi></math></span> and Spearman’s <span><math><mi>ρ</mi></math></span> and derived a sharp lower, as well as a non-sharp upper bound for <span><math><mi>ρ</mi></math></span> given <span><math><mi>ϕ</mi></math></span>. Considering that the proofs for establishing these inequalities are novel and interesting, but technically quite involved we here provide alternative simpler proofs mainly building upon shuffles, symmetry, denseness and mass shifting. As a by-product of these proofs we derive several additional results on shuffle rearrangements and the interplay between diagonal copulas and shuffles which are of independent interest. Moreover we finally show that we can get closer to the (non-sharp) upper bound than established in the literature so far.</p></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"457 ","pages":"Article 116259"},"PeriodicalIF":2.1000,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0377042724005089/pdfft?md5=496684547289907e38a430b957fc4235&pid=1-s2.0-S0377042724005089-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Revisiting the region determined by Spearman’s ρ and Spearman’s footrule ϕ\",\"authors\":\"Marco Tschimpke , Manuela Schreyer , Wolfgang Trutschnig\",\"doi\":\"10.1016/j.cam.2024.116259\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Kokol and Stopar (2023) recently studied the exact region <span><math><msub><mrow><mi>Ω</mi></mrow><mrow><mi>ϕ</mi><mo>,</mo><mi>ρ</mi></mrow></msub></math></span> determined by Spearman’s footrule <span><math><mi>ϕ</mi></math></span> and Spearman’s <span><math><mi>ρ</mi></math></span> and derived a sharp lower, as well as a non-sharp upper bound for <span><math><mi>ρ</mi></math></span> given <span><math><mi>ϕ</mi></math></span>. Considering that the proofs for establishing these inequalities are novel and interesting, but technically quite involved we here provide alternative simpler proofs mainly building upon shuffles, symmetry, denseness and mass shifting. As a by-product of these proofs we derive several additional results on shuffle rearrangements and the interplay between diagonal copulas and shuffles which are of independent interest. Moreover we finally show that we can get closer to the (non-sharp) upper bound than established in the literature so far.</p></div>\",\"PeriodicalId\":50226,\"journal\":{\"name\":\"Journal of Computational and Applied Mathematics\",\"volume\":\"457 \",\"pages\":\"Article 116259\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-09-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0377042724005089/pdfft?md5=496684547289907e38a430b957fc4235&pid=1-s2.0-S0377042724005089-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Computational and Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0377042724005089\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0377042724005089","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Revisiting the region determined by Spearman’s ρ and Spearman’s footrule ϕ
Kokol and Stopar (2023) recently studied the exact region determined by Spearman’s footrule and Spearman’s and derived a sharp lower, as well as a non-sharp upper bound for given . Considering that the proofs for establishing these inequalities are novel and interesting, but technically quite involved we here provide alternative simpler proofs mainly building upon shuffles, symmetry, denseness and mass shifting. As a by-product of these proofs we derive several additional results on shuffle rearrangements and the interplay between diagonal copulas and shuffles which are of independent interest. Moreover we finally show that we can get closer to the (non-sharp) upper bound than established in the literature so far.
期刊介绍:
The Journal of Computational and Applied Mathematics publishes original papers of high scientific value in all areas of computational and applied mathematics. The main interest of the Journal is in papers that describe and analyze new computational techniques for solving scientific or engineering problems. Also the improved analysis, including the effectiveness and applicability, of existing methods and algorithms is of importance. The computational efficiency (e.g. the convergence, stability, accuracy, ...) should be proved and illustrated by nontrivial numerical examples. Papers describing only variants of existing methods, without adding significant new computational properties are not of interest.
The audience consists of: applied mathematicians, numerical analysts, computational scientists and engineers.