{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T20:43:06Z","timestamp":1769632986257,"version":"3.49.0"},"reference-count":26,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,7]],"date-time":"2021-12-07T00:00:00Z","timestamp":1638835200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A new, flexible claim-size Chen density is derived for modeling asymmetric data (negative and positive) with different types of kurtosis (leptokurtic, mesokurtic and platykurtic). The new function is used for modeling bimodal asymmetric medical data, water resource bimodal asymmetric data and asymmetric negatively skewed insurance-claims payment triangle data. The new density accommodates the \u201csymmetric\u201d, \u201cunimodal right skewed\u201d, \u201cunimodal left skewed\u201d, \u201cbimodal right skewed\u201d and \u201cbimodal left skewed\u201d densities. The new hazard function can be \u201cdecreasing\u2013constant\u2013increasing (bathtub)\u201d, \u201cmonotonically increasing\u201d, \u201cupside down constant\u2013increasing\u201d, \u201cmonotonically decreasing\u201d, \u201cJ shape\u201d and \u201cupside down\u201d. Four risk indicators are analyzed under insurance-claims payment triangle data using the proposed distribution. Since the insurance-claims data are a quarterly time series, we analyzed them using the autoregressive regression model AR(1). Future insurance-claims forecasting is very important for insurance companies to avoid uncertainty about big losses that may be produced from future claims.<\/jats:p>","DOI":"10.3390\/sym13122357","type":"journal-article","created":{"date-parts":[[2021,12,7]],"date-time":"2021-12-07T20:23:21Z","timestamp":1638908601000},"page":"2357","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Asymmetric Density for Risk Claim-Size Data: Prediction and Bimodal Data Applications"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3456-8393","authenticated-orcid":false,"given":"Mansour","family":"Shrahili","sequence":"first","affiliation":[{"name":"Department of Statistics and Operations Research, King Saud University, Riyadh 11451, Saudi Arabia"}]},{"given":"Ibrahim","family":"Elbatal","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4589-4944","authenticated-orcid":false,"given":"Haitham","family":"M. Yousof","sequence":"additional","affiliation":[{"name":"Department of Statistics, Mathematics and Insurance, Benha University, Benha 13513, Egypt"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Klugman, S.A., Panjer, H.H., and Willmot, G.E. (2012). Loss Models: From Data to Decisions, John Wiley & Sons.","DOI":"10.1002\/9781118787106"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1080\/10920277.1999.10595804","article-title":"Raising Value at Risk","volume":"3","author":"Wirch","year":"1999","journal-title":"N. Am. Actuar. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1519","DOI":"10.1016\/S0378-4266(02)00272-8","article-title":"Expected Shortfall and Beyond","volume":"26","author":"Tasche","year":"2002","journal-title":"J. Bank. 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