{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T10:36:15Z","timestamp":1780050975960,"version":"3.53.1"},"reference-count":29,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,13]],"date-time":"2023-07-13T00:00:00Z","timestamp":1689206400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"JSPS KAKENHI","doi-asserted-by":"publisher","award":["JP19K23226"],"award-info":[{"award-number":["JP19K23226"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"JSPS KAKENHI","doi-asserted-by":"publisher","award":["JP21K01581"],"award-info":[{"award-number":["JP21K01581"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"JSPS KAKENHI","doi-asserted-by":"publisher","award":["JP21K13275"],"award-info":[{"award-number":["JP21K13275"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"JSPS KAKENHI","doi-asserted-by":"publisher","award":["JP20K03759"],"award-info":[{"award-number":["JP20K03759"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Asymmetry in the upper and lower tails is an important feature in modeling bivariate distributions. This article focuses on the log ratio between the tail probabilities at upper and lower corners as a measure of tail asymmetry. Asymptotic behavior of this measure at extremely large and small thresholds is explored with particular emphasis on the skew-normal copula. Our numerical studies reveal that, when the correlation or skewness parameters are around at the boundary values, some asymptotic tail approximations of the skew-normal copulas proposed in the literature are not suitable to compute the measure of tail asymmetry with practically extremal thresholds.<\/jats:p>","DOI":"10.3390\/sym15071410","type":"journal-article","created":{"date-parts":[[2023,7,14]],"date-time":"2023-07-14T00:28:06Z","timestamp":1689294486000},"page":"1410","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["On a Measure of Tail Asymmetry for the Bivariate Skew-Normal Copula"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4447-6663","authenticated-orcid":false,"given":"Toshinao","family":"Yoshiba","sequence":"first","affiliation":[{"name":"Graduate School of Management, Tokyo Metropolitan University, 18F 1-4-1 Marunouchi, Chiyoda-ku 100-0005, Japan"},{"name":"Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa 190-8562, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7940-1418","authenticated-orcid":false,"given":"Takaaki","family":"Koike","sequence":"additional","affiliation":[{"name":"Graduate School of Economics, Hitotsubashi University, 2-1, Naka, Kunitachi 186-8601, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2648-6769","authenticated-orcid":false,"given":"Shogo","family":"Kato","sequence":"additional","affiliation":[{"name":"Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa 190-8562, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1016\/S0304-405X(02)00068-5","article-title":"Asymmetric correlations of equity portfolios","volume":"63","author":"Ang","year":"2002","journal-title":"J. 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