{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:49:00Z","timestamp":1760240940864,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2019,10,24]],"date-time":"2019-10-24T00:00:00Z","timestamp":1571875200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11771343","11901184"],"award-info":[{"award-number":["11771343","11901184"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["531107051210"],"award-info":[{"award-number":["531107051210"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Basic Research Plan in Shaanxi Province of China","award":["2014JQ2-1003"],"award-info":[{"award-number":["2014JQ2-1003"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The aim of this paper is to construct two new classes of multivariate risk statistics, and to study their properties. We, first, introduce the multivariate shortfall risk statistics and multivariate divergence risk statistics. Then, their basic properties are studied, and their representation results are provided. Furthermore, their coherency is also characterized by means of the corresponding loss function. Finally, entropic risk statistics are given to illustrate the proposed new classes of multivariate risk statistics. The relationship between multivariate shortfall and divergence risk statistics is also discussed.<\/jats:p>","DOI":"10.3390\/e21111031","type":"journal-article","created":{"date-parts":[[2019,10,25]],"date-time":"2019-10-25T04:41:27Z","timestamp":1571978487000},"page":"1031","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Multivariate Shortfall and Divergence Risk Statistics"],"prefix":"10.3390","volume":"21","author":[{"given":"Haiyan","family":"Song","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7362-1666","authenticated-orcid":false,"given":"Xianfu","family":"Zeng","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanhong","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Finance and Statistics, Hunan University, Changsha 410082, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yijun","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1111\/1467-9965.00068","article-title":"Coherent measures of risk","volume":"9","author":"Artzner","year":"1999","journal-title":"Math. Financ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1007\/s007800200072","article-title":"Convex measures of risk and trading constraints","volume":"6","author":"Schied","year":"2002","journal-title":"Financ. Stoch."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1473","DOI":"10.1016\/S0378-4266(02)00270-4","article-title":"Putting order in risk measures","volume":"26","author":"Frittelli","year":"2002","journal-title":"J. Bank. Financ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/s00780-006-0024-2","article-title":"Optimal investments for risk- and ambiguity-averse preferences: A duality approach","volume":"11","author":"Schied","year":"2007","journal-title":"Financ. Stoch."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1111\/j.1467-9965.2007.00311.x","article-title":"An old-new concept of convex risk measures: The optimized certainty equivalent","volume":"17","author":"Teboulle","year":"2007","journal-title":"Math. Financ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1111\/j.1467-9965.2006.00277.x","article-title":"Distribution-invariant risk measures, information, and dynamic consistency","volume":"16","author":"Weber","year":"2006","journal-title":"Math. Financ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1007\/s00780-015-0279-6","article-title":"Risk measures with the CxLS property","volume":"20","author":"Delbaen","year":"2016","journal-title":"Financ. Stoch."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1750026","DOI":"10.1142\/S0219024917500261","article-title":"Set-valued shortfall and divergence risk measures","volume":"20","author":"Ararat","year":"2017","journal-title":"Int. J. Theor. Appl. Financ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1137\/16M1087357","article-title":"Multivariate shortfall risk allocation and systemic risk","volume":"9","author":"Armenti","year":"2018","journal-title":"SIAM J. Financ. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1287\/moor.12.2.224","article-title":"Penalty functions and duality in stochastic programming via \u03d5-divergence functionals","volume":"12","author":"Teboulle","year":"1987","journal-title":"Math. Oper. Res."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Xu, M., and Angulo, J. (2019). Divergence-based risk measures: A discussion on sensitivities and extensions. Entropy, 21.","DOI":"10.3390\/e21070634"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.insmatheco.2005.08.008","article-title":"Consistent risk measures for portfolio vectors","volume":"38","author":"Burgert","year":"2006","journal-title":"Insur. Math. Econ."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"R\u00fcschendorf, L. (2013). Mathematical Risk Analysis, Springer.","DOI":"10.1007\/978-3-642-33590-7"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1524\/stnd.2011.1099","article-title":"Law invariant risk measures on L\u221e(Rd)","volume":"28","author":"Ekeland","year":"2011","journal-title":"Stat. Risk. Model."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1111\/j.1467-9965.2010.00453.x","article-title":"Comonotonic measures of multivariate risks","volume":"22","author":"Ekeland","year":"2012","journal-title":"Math. Financ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.spl.2014.03.005","article-title":"Coherent and convex risk measures for portfolios and applications","volume":"90","author":"Wei","year":"2014","journal-title":"Stat. Probab. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1007\/s11117-017-0517-6","article-title":"Coherent and convex loss-based risk measures for portfolio vectors","volume":"22","author":"Chen","year":"2018","journal-title":"Positivity"},{"key":"ref_18","unstructured":"Heyde, C.C., Kou, S., and Peng, X. (2006). What Is a Good Risk Measure: Bridging the Gaps between Data, Coherent Risk Measures, and Insurance Risk Measures, Columbia University. Available online: https:\/\/www.math.ust.hk\/~maxhpeng\/KOU_hkpv1.pdf."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1287\/moor.1120.0577","article-title":"External risk measures and Basel Accords","volume":"38","author":"Kou","year":"2013","journal-title":"Math. Oper. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s11579-014-0136-y","article-title":"Quasiconvex risk statistics with scenario analysis","volume":"9","author":"Tian","year":"2015","journal-title":"Math. Financ. Econ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1016\/j.orl.2012.09.011","article-title":"A note on convex risk statistic","volume":"40","author":"Tian","year":"2012","journal-title":"Oper. Res. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.spl.2017.08.004","article-title":"Set-valued risk statistics with scenario analysis","volume":"131","author":"Chen","year":"2017","journal-title":"Stat. Probab. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"5585","DOI":"10.1080\/03610926.2018.1515957","article-title":"Multivariate convex risk statistics with scenario analysis","volume":"48","author":"Liu","year":"2019","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"F\u00f6llmer, H., and Schied, A. (2011). Stochastic Finance: An Introduction in Discrete Time, Walter de Gruyter. [3rd ed.].","DOI":"10.1515\/9783110218053"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/11\/1031\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:28:58Z","timestamp":1760189338000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/11\/1031"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,24]]},"references-count":24,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2019,11]]}},"alternative-id":["e21111031"],"URL":"https:\/\/doi.org\/10.3390\/e21111031","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2019,10,24]]}}}