{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:27:40Z","timestamp":1760243260590,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2014,6,12]],"date-time":"2014-06-12T00:00:00Z","timestamp":1402531200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The maximum entropy method was originally proposed as a variational technique to determine probability densities from the knowledge of a few expected values. The applications of the method beyond its original role in statistical physics are manifold. An interesting feature of the method is its potential to incorporate errors in the data. Here, we examine two possible ways of doing that. The two approaches have different intuitive interpretations, and one of them allows for error estimation. Our motivating example comes from the field of risk analysis, but the statement of the problem might as well come from any branch of applied sciences. We apply the methodology to a problem consisting of the determination of a probability density from a few values of its numerically-determined Laplace transform. This problem can be mapped onto a problem consisting of the determination of a probability density on [0, 1] from the knowledge of a few of its fractional moments up to some measurement errors stemming from insufficient data.<\/jats:p>","DOI":"10.3390\/e16063257","type":"journal-article","created":{"date-parts":[[2014,6,12]],"date-time":"2014-06-12T11:14:31Z","timestamp":1402571671000},"page":"3257-3272","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Density Reconstructions with Errors in the Data"],"prefix":"10.3390","volume":"16","author":[{"given":"Erika","family":"Gomes-Gon\u00e7alves","sequence":"first","affiliation":[{"name":"Business Administration, Univ. Carlos III de Madrid, Calle Madrid 126, 28093 Getafe, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Henryk","family":"Gzyl","sequence":"additional","affiliation":[{"name":"Centro de Finanzas, IESA, Ave. Iesa San Bernardino, Caracas 1010, Venezuela"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Silvia","family":"Mayoral","sequence":"additional","affiliation":[{"name":"Business Administration, Univ. Carlos III de Madrid, Calle Madrid 126, 28093 Getafe, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2014,6,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1017\/S0334270000006901","article-title":"On Lognormal Random Variables: I The Characteristic Function","volume":"32","author":"Leipnik","year":"1991","journal-title":"J. Aust. Math. Soc. Series B"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Panjer, H. (2006). Operational Risk: Modeling and Analytics, John Wiley & Sons.","DOI":"10.1002\/0470051310"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3","DOI":"10.21314\/JOP.2013.117","article-title":"A comparison of numerical approaches to determine the severity of losses","volume":"8","author":"Gzyl","year":"2013","journal-title":"J. Oper. Risk"},{"key":"ref_4","first-page":"447","article-title":"On minimization and maximization of entropy functionals in various disciplines","volume":"17","author":"Cherny","year":"2003","journal-title":"Theory Probab. Appl"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"148","DOI":"10.1214\/aop\/1176996454","article-title":"I-divergence geometry of probability distributions and minimization problems","volume":"3","author":"Csiszar","year":"1975","journal-title":"Ann. Probab"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1214\/aop\/1176993227","article-title":"Generalized I-projection and a conditional limit theorem","volume":"12","author":"Csiszar","year":"1984","journal-title":"Ann. Probab"},{"key":"ref_7","unstructured":"Kapur, N. (1996). Maximum Entropy Models in Science and Engineering, Wiley Interscience."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Gzyl, H., and Vel\u00e1squez, Y. (2011). Linear Inverse Problems: The Maximum Entropy Connection, World Scientific.","DOI":"10.1142\/9789814338783"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.insmatheco.2010.03.011","article-title":"A method for determining risk aversion functions from uncertain market prices of risk","volume":"47","author":"Gzyl","year":"2010","journal-title":"Insur. Math. Econ"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1103\/PhysRev.106.620","article-title":"Information theory and statistical physics","volume":"106","author":"Jaynes","year":"1957","journal-title":"Phys. Rev"},{"key":"ref_11","first-page":"425","article-title":"Minimisation de l\u2019information de Kullback et minimisation de l\u2019entropie sous une contrainte quadratique","volume":"306","author":"Gamboa","year":"1988","journal-title":"CRAS Paris, S\u00e8rie I Math"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Borwein, J.M., and Lewis, A.S. (2000). Convex Analysis and Nonlinear Optimization, Springer Verlag.","DOI":"10.1007\/978-1-4757-9859-3"},{"key":"ref_13","unstructured":"Gomes, E.(Univ. Carlos III de Madrid, Getafe, Spain); Gzyl, H. (IESA, Caracas, Venezuela); Mayoral, S. (Univ. Carlos III de Madrid, Getafe, Spain). A maxentropic approach to determine operational risk losses. In preparation, 2014."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/6\/3257\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:12:24Z","timestamp":1760217144000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/6\/3257"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,6,12]]},"references-count":13,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2014,6]]}},"alternative-id":["e16063257"],"URL":"https:\/\/doi.org\/10.3390\/e16063257","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2014,6,12]]}}}