{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:27:09Z","timestamp":1760243229133,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2014,4,17]],"date-time":"2014-04-17T00:00:00Z","timestamp":1397692800000},"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) criterion has been successfully used in fields such as parameter estimation, system identification and the supervised machine learning. There is in general no explicit expression for the optimal MEE estimate unless some constraints on the conditional distribution are imposed. A recent paper has proved that if the conditional density is conditionally symmetric and unimodal (CSUM), then the optimal MEE estimate (with Shannon entropy) equals the conditional median. In this study, we extend this result to the generalized MEE estimation where the optimality criterion is the Renyi entropy or equivalently, the \u03b1-order information potential (IP).<\/jats:p>","DOI":"10.3390\/e16042223","type":"journal-article","created":{"date-parts":[[2014,4,17]],"date-time":"2014-04-17T12:21:27Z","timestamp":1397737287000},"page":"2223-2233","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An Extended Result on the Optimal Estimation Under the Minimum Error Entropy Criterion"],"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":[[2014,4,17]]},"reference":[{"key":"ref_1","unstructured":"Haykin, S. (1996). Adaptive Filtering Theory, Prentice Hall."},{"key":"ref_2","unstructured":"Kailath, T., Sayed, A.H., and Hassibi, B. (2000). Linear Estimation, Prentice Hall."},{"key":"ref_3","unstructured":"Papoulis, A., and Pillai, S.U. (2002). Probability, Random Variables, and Stochastic Processes, McGraw-Hill Education."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"986","DOI":"10.1109\/5.231338","article-title":"Signal processing with fractional lower order moments: Stable processes and their applications","volume":"81","author":"Shao","year":"1993","journal-title":"IEEE Proc"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1540","DOI":"10.1109\/49.339922","article-title":"Least mean p-power error criterion for adaptive FIR filter","volume":"12","author":"Pei","year":"1994","journal-title":"IEEE J. Sel. Areas Commun"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Rousseeuw, P.J., and Leroy, A.M. (1987). Robust Regression and Outlier Detection, John Wiley & Sons.","DOI":"10.1002\/0471725382"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1109\/9.989082","article-title":"Robustness and risk-sensitive filtering","volume":"47","author":"Boel","year":"2002","journal-title":"IEEE Trans. Automat. Control"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1109\/18.79934","article-title":"On optimal estimation with respect to a large family of cost functions","volume":"37","author":"Hall","year":"1991","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_9","unstructured":"Cover, T.M., and Thomas, J.A. (1991). Element of Information Theory, Wiley & Son, Inc."},{"key":"ref_10","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 Trans. Inform. Theory"},{"key":"ref_11","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":"Inf. Control"},{"key":"ref_12","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":"Inf. Sci"},{"key":"ref_13","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. Inf. Theory"},{"key":"ref_14","unstructured":"Renyi, A. (July,, January 20). On Measures of Entropy and Information. Berkeley, CA, USA."},{"key":"ref_15","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_16","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_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. Neur. Netw"},{"key":"ref_19","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_20","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_21","doi-asserted-by":"crossref","first-page":"941","DOI":"10.1007\/s00034-007-9004-9","article-title":"Stochastic gradient algorithm under (h, \u03c6)-entropy criterion","volume":"26","author":"Chen","year":"2007","journal-title":"Circuits Syst. Signal Process"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1109\/TNN.2010.2050212","article-title":"Mean-square convergence analysis of ADALINE training with minimum error entropy criterion","volume":"21","author":"Chen","year":"2010","journal-title":"IEEE Trans. Neur. Netw"},{"key":"ref_23","unstructured":"Hardy, G.H., Littlewood, J.E., and Polya, G. (1934). Inequalities, Cambridge University Press."},{"key":"ref_24","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"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/4\/2223\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:10:27Z","timestamp":1760217027000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/16\/4\/2223"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,4,17]]},"references-count":24,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2014,4]]}},"alternative-id":["e16042223"],"URL":"https:\/\/doi.org\/10.3390\/e16042223","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2014,4,17]]}}}