{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:41:16Z","timestamp":1760218876264,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2014,2,10]],"date-time":"2014-02-10T00:00:00Z","timestamp":1391990400000},"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 minimum error entropy (MEE) estimation is concerned with the estimation of a certain random variable (unknown variable) based on another random variable (observation), so that the entropy of the estimation error is minimized. This estimation method may outperform the well-known minimum mean square error (MMSE) estimation especially for non-Gaussian situations. There is an important performance bound on the MEE estimation, namely the W-S lower bound, which is computed as the conditional entropy of the unknown variable given observation. Though it has been known in the literature for a considerable time, up to now there is little study on this performance bound. In this paper, we reexamine the W-S lower bound. Some basic properties of the W-S lower bound are presented, and the characterization of Gaussian distribution using the W-S lower bound is investigated.<\/jats:p>","DOI":"10.3390\/e16020814","type":"journal-article","created":{"date-parts":[[2014,2,10]],"date-time":"2014-02-10T11:52:37Z","timestamp":1392033157000},"page":"814-824","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Note on the W-S Lower Bound of the MEE Estimation"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1710-3818","authenticated-orcid":false,"given":"Badong","family":"Chen","sequence":"first","affiliation":[{"name":"Institute of Artificial Intelligence and Robotics, Xi'an Jiaotong University, Xi'an 710049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guangmin","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute of Artificial Intelligence and Robotics, Xi'an Jiaotong University, Xi'an 710049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nanning","family":"Zheng","sequence":"additional","affiliation":[{"name":"Institute of Artificial Intelligence and Robotics, Xi'an Jiaotong University, Xi'an 710049, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jose","family":"Principe","sequence":"additional","affiliation":[{"name":"Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2014,2,10]]},"reference":[{"key":"ref_1","unstructured":"Kailath, T., Sayed, A.H., and Hassibi, B (2000). Linear Estimation;, Prentice Hall."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"264","DOI":"10.1109\/TIT.1970.1054444","article-title":"Entropy analysis of estimating systems","volume":"16","author":"Weidemann","year":"1970","journal-title":"IEEE T. Inform. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/S0019-9958(76)90140-6","article-title":"An application of the information theory to estimation problems","volume":"32","author":"Tomita","year":"1976","journal-title":"Inform. Contr"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0020-0255(79)90039-2","article-title":"Linear prediction, filtering and smoothing: An information theoretic approach","volume":"17","author":"Kalata","year":"1979","journal-title":"Inform. Sci"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/S0019-9958(82)91153-6","article-title":"An extension of the entropy theorem for parameter estimation","volume":"53","author":"Minamide","year":"1982","journal-title":"Inform. Contr"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/0020-0255(94)90043-4","article-title":"Minimum entropy of error principle in estimation","volume":"79","author":"Janzura","year":"1994","journal-title":"Inform. Sci"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"937","DOI":"10.1016\/j.sigpro.2004.11.028","article-title":"Minimum-entropy estimation in semi-parametric models","volume":"85","author":"Wolsztynski","year":"2005","journal-title":"Signal Process"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Principe, J.C. (2010). Information Theoretic Learning: Renyi\u2019s Entropy and Kernel Perspectives;, Springer.","DOI":"10.1007\/978-1-4419-1570-2"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Chen, B., Zhu, Y., Hu, J., and Principe, J.C. (2013). System Parameter Identification: Information Criteria and Algorithms;, Elsevier Inc.","DOI":"10.1016\/B978-0-12-404574-3.00005-1"},{"key":"ref_10","unstructured":"Cover, T.M., and Thomas, J.A. (1991). Element of Information Theory;, Wiley & Sons Inc."},{"key":"ref_11","first-page":"331","article-title":"Minimum entropy of error estimate for multi-dimensional parameter and finite-state-space observations","volume":"31","author":"Otahal","year":"1995","journal-title":"Kybernetika"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3166","DOI":"10.1109\/TIT.2008.924686","article-title":"On the minimum entropy of a mixture of unimodal and symmetric distributions","volume":"54","author":"Chen","year":"2008","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1016\/j.jfranklin.2009.11.009","article-title":"On optimal estimations with minimum error entropy criterion","volume":"347","author":"Chen","year":"2010","journal-title":"J. Franklin Inst"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3313","DOI":"10.1016\/j.sigpro.2010.05.028","article-title":"A new interpretation on the MMSE as a robust MEE criterion","volume":"90","author":"Chen","year":"2010","journal-title":"Signal Process"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"966","DOI":"10.3390\/e14050966","article-title":"Some further results on the minimum error entropy estimation","volume":"14","author":"Chen","year":"2012","journal-title":"Entropy"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2311","DOI":"10.3390\/e14112311","article-title":"On the smoothed minimum error entropy criterion","volume":"14","author":"Chen","year":"2012","journal-title":"Entropy"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1780","DOI":"10.1109\/TSP.2002.1011217","article-title":"An error-entropy minimization algorithm for supervised training of nonlinear adaptive systems","volume":"50","author":"Erdogmus","year":"2002","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1035","DOI":"10.1109\/TNN.2002.1031936","article-title":"Generalized information potential criterion for adaptive system training","volume":"13","author":"Erdogmus","year":"2002","journal-title":"IEEE Trans. Neural Network"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1966","DOI":"10.1109\/TSP.2003.812843","article-title":"Convergence properties and data efficiency of the minimum error entropy criterion in Adaline training","volume":"51","author":"Erdogmus","year":"2003","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1109\/SP-M.2006.248709","article-title":"From linear adaptive filtering to nonlinear information processing\u2014The design and analysis of information processing systems","volume":"23","author":"Erdogmus","year":"2006","journal-title":"IEEE Signal Process. Mag"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1184","DOI":"10.1109\/78.995074","article-title":"Entropy minimization for supervised digital communications channel equalization","volume":"50","author":"Santamaria","year":"2002","journal-title":"IEEE Trans. Signal Process"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1007\/s00034-007-9004-9","article-title":"Stochastic gradient algorithm under (h, \u03d5)-entropy criterion","volume":"26","author":"Chen","year":"2007","journal-title":"Circuits Syst. Signal Process"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1214\/aoms\/1177704579","article-title":"A characterization of the multivariate normal distribution","volume":"33","author":"Ghurye","year":"1962","journal-title":"Ann. Math. Stat"},{"key":"ref_24","first-page":"15","article-title":"Characterizing the general multivariate normal distribution through the conditional distributions","volume":"12","author":"Fidalgo","year":"1997","journal-title":"Extr. Math"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/BF02613616","article-title":"Characterization of the multivariate normal distribution by conditional normal distributions","volume":"38","author":"Bischoff","year":"1991","journal-title":"Metrika"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"486","DOI":"10.1214\/aoms\/1177697088","article-title":"A note on a characterization of the multivariate normal distribution","volume":"41","author":"Fisk","year":"1970","journal-title":"Ann. Math. Stat"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/BF02613618","article-title":"On a recent characterization of the bivariate normal distribution","volume":"38","author":"Hamedani","year":"1991","journal-title":"Metrika"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/2\/814\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:08:02Z","timestamp":1760216882000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/2\/814"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,2,10]]},"references-count":27,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2014,2]]}},"alternative-id":["e16020814"],"URL":"https:\/\/doi.org\/10.3390\/e16020814","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2014,2,10]]}}}