{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T15:57:36Z","timestamp":1780675056243,"version":"3.54.1"},"reference-count":113,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2018,5,6]],"date-time":"2018-05-06T00:00:00Z","timestamp":1525564800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Department of Science and Technology, Government of India","award":["INSPIRE Faculty Research Grant"],"award-info":[{"award-number":["INSPIRE Faculty Research Grant"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Entropy and relative entropy measures play a crucial role in mathematical information theory. The relative entropies are also widely used in statistics under the name of divergence measures which link these two fields of science through the minimum divergence principle. Divergence measures are popular among statisticians as many of the corresponding minimum divergence methods lead to robust inference in the presence of outliers in the observed data; examples include the   \u03d5   -divergence, the density power divergence, the logarithmic density power divergence and the recently developed family of logarithmic super divergence (LSD). In this paper, we will present an alternative information theoretic formulation of the LSD measures as a two-parameter generalization of the relative   \u03b1   -entropy, which we refer to as the general    ( \u03b1 , \u03b2 )    -entropy. We explore its relation with various other entropies and divergences, which also generates a two-parameter extension of Renyi entropy measure as a by-product. This paper is primarily focused on the geometric properties of the relative    ( \u03b1 , \u03b2 )    -entropy or the LSD measures; we prove their continuity and convexity in both the arguments along with an extended Pythagorean relation under a power-transformation of the domain space. We also derive a set of sufficient conditions under which the forward and the reverse projections of the relative    ( \u03b1 , \u03b2 )    -entropy exist and are unique. Finally, we briefly discuss the potential applications of the relative    ( \u03b1 , \u03b2 )    -entropy or the LSD measures in statistical inference, in particular, for robust parameter estimation and hypothesis testing. Our results on the reverse projection of the relative    ( \u03b1 , \u03b2 )    -entropy establish, for the first time, the existence and uniqueness of the minimum LSD estimators. Numerical illustrations are also provided for the problem of estimating the binomial parameter.<\/jats:p>","DOI":"10.3390\/e20050347","type":"journal-article","created":{"date-parts":[[2018,5,7]],"date-time":"2018-05-07T03:12:21Z","timestamp":1525662741000},"page":"347","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["A Generalized Relative (\u03b1, \u03b2)-Entropy: Geometric Properties and Applications to Robust Statistical Inference"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3688-4584","authenticated-orcid":false,"given":"Abhik","family":"Ghosh","sequence":"first","affiliation":[{"name":"Indian Statistical Institute, Kolkata 700108, India"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ayanendranath","family":"Basu","sequence":"additional","affiliation":[{"name":"Indian Statistical Institute, Kolkata 700108, India"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2018,5,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1109\/JRPROC.1949.232969","article-title":"Communication in the presence of noise","volume":"37","author":"Shannon","year":"1949","journal-title":"Proc. IRE"},{"key":"ref_3","unstructured":"Shannon, C.E., and Weaver, W. (1949). The Mathematical Theory of Communication, University of Illinois Press."},{"key":"ref_4","first-page":"3","article-title":"The entropy concept in probability theory","volume":"8","author":"Khinchin","year":"1953","journal-title":"Uspekhi Matematicheskikh Nauk"},{"key":"ref_5","first-page":"17","article-title":"On the fundamental theorems of information theory","volume":"11","author":"Khinchin","year":"1956","journal-title":"Uspekhi Matematicheskikh Nauk"},{"key":"ref_6","unstructured":"Khinchin, A.I. (1957). Mathematical Foundations of Information Theory, Dover Publications."},{"key":"ref_7","unstructured":"Kolmogorov, A.N. (1950). Foundations of the Theory of Probability, Chelsea Publishing Co."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1109\/TIT.1956.1056823","article-title":"On the Shannon theory of information transmission in the case of continuous signals","volume":"IT-2","author":"Kolmogorov","year":"1956","journal-title":"IRE Trans. Inf. Theory"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1214\/aoms\/1177729487","article-title":"An application of information theory to multivariate analysis","volume":"23","author":"Kullback","year":"1952","journal-title":"Ann. Math. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1063\/1.1721117","article-title":"A note on information theory","volume":"24","author":"Kullback","year":"1953","journal-title":"J. Appl. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1214\/aoms\/1177728660","article-title":"Certain inequalities in information theory and the Cramer-Rao inequality","volume":"25","author":"Kullback","year":"1954","journal-title":"Ann. Math. Stat."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1214\/aoms\/1177728353","article-title":"An application of information theory to multivariate analysis II","volume":"27","author":"Kullback","year":"1956","journal-title":"Ann. Math. Stat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Rosenkrantz, R.D. (1983). E T Jaynes: Papers on Probability, Statistics and Statistical Physics, Springer Science and Business Media.","DOI":"10.1007\/978-94-009-6581-2"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1109\/TIT.1981.1056374","article-title":"Maximum entropy and conditional probability","volume":"27","author":"Cover","year":"1981","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5063","DOI":"10.1109\/TIT.2015.2449311","article-title":"Minimization Problems Based on Relative \u03b1-Entropy I: Forward Projection","volume":"61","author":"Kumar","year":"2015","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1109\/TIT.2006.887466","article-title":"Guessing under source uncertainty","volume":"53","author":"Sundaresan","year":"2007","journal-title":"Proc. IEEE Trans. Inf. Theory"},{"key":"ref_18","first-page":"146","article-title":"I-divergence geometry of probability distributions and minimization problems","volume":"3","year":"1975","journal-title":"Ann. Probab."},{"key":"ref_19","first-page":"768","article-title":"Sanov property, generalized I -projection, and a conditional limit theorem","volume":"12","year":"1984","journal-title":"Ann. Probab."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Csisz\u00e1r, I., and Shields, P. (2004). Information Theory and Statistics: A Tutorial, NOW Publishers.","DOI":"10.1561\/9781933019543"},{"key":"ref_21","first-page":"205","article-title":"Information geometry and alternating minimization procedures","volume":"1","author":"Tusnady","year":"1984","journal-title":"Stat. Decis."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Amari, S.I., Karakida, R., and Oizumi, M. (arXiv, 2017). Information Geometry Connecting Wasserstein Distance and Kullback-Leibler Divergence via the Entropy-Relaxed Transportation Problem, arXiv.","DOI":"10.1007\/s41884-018-0002-8"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.dam.2014.10.004","article-title":"Fisher information distance: A geometrical reading","volume":"197","author":"Costa","year":"2015","journal-title":"Discret. Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1543","DOI":"10.1109\/LSP.2016.2606661","article-title":"Guaranteed bounds on the Kullback-Leibler divergence of univariate mixtures","volume":"23","author":"Nielsen","year":"2016","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_25","first-page":"183","article-title":"Information geometry of divergence functions","volume":"58","author":"Amari","year":"2010","journal-title":"Bull. Pol. Acad. Sci. Tech. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1606","DOI":"10.3390\/e14091606","article-title":"Kullback-Leibler divergence measure for multivariate skew-normal distributions","volume":"14","year":"2012","journal-title":"Entropy"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"5455","DOI":"10.1109\/TIT.2011.2159046","article-title":"The Burbea-Rao and Bhattacharyya Centroids","volume":"57","author":"Nielsen","year":"2011","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4091","DOI":"10.1137\/140962802","article-title":"Kullback\u2013Leibler approximation for probability measures on infinite dimensional spaces","volume":"47","author":"Pinski","year":"2015","journal-title":"SIAM J. Math. Anal."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1287\/moor.1100.0449","article-title":"Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-Lojasiewicz inequality","volume":"35","author":"Attouch","year":"2010","journal-title":"Math. Oper. Res."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3023","DOI":"10.1016\/j.physa.2010.03.045","article-title":"Maximization of statistical heterogeneity: From Shannon\u2019s entropy to Gini\u2019s index","volume":"389","author":"Eliazar","year":"2010","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"P03008","DOI":"10.1088\/1742-5468\/2011\/03\/P03008","article-title":"Non-equilibrium steady states: Maximization of the Shannon entropy associated with the distribution of dynamical trajectories in the presence of constraints","volume":"2011","author":"Monthus","year":"2011","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1016\/j.neucom.2013.12.018","article-title":"Application of wavelet energy and Shannon entropy for feature extraction in gearbox fault detection under varying speed conditions","volume":"133","author":"Bafroui","year":"2014","journal-title":"Neurocomputing"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1111\/j.1538-4632.2010.00800.x","article-title":"Space, Scale, and Scaling in Entropy Maximizing","volume":"42","author":"Batty","year":"2010","journal-title":"Geogr. Anal."},{"key":"ref_34","unstructured":"Oikonomou, T., and Bagci, G.B. (arXiv, 2018). Entropy Maximization with Linear Constraints: The Uniqueness of the Shannon Entropy, arXiv."},{"key":"ref_35","unstructured":"Hoang, D.T., Song, J., Periwal, V., and Jo, J. (arXiv, 2017). Maximizing weighted Shannon entropy for network inference with little data, arXiv."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"044304","DOI":"10.1063\/1.4994922","article-title":"Characteristic features of the Shannon information entropy of dipolar Bose-Einstein condensates","volume":"147","author":"Sriraman","year":"2017","journal-title":"J. Chem. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1233","DOI":"10.1109\/TASLP.2015.2427520","article-title":"Speech enhancement under low SNR conditions via noise estimation using sparse and low-rank NMF with Kullback-Leibler divergence","volume":"23","author":"Sun","year":"2015","journal-title":"IEEE Trans. Audio Speech Lang. Process."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"5812","DOI":"10.1109\/TSP.2015.2468677","article-title":"Derivation of the PHD and CPHD Filters Based on Direct Kullback-Leibler Divergence Minimization","volume":"63","author":"Vo","year":"2015","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1770","DOI":"10.1109\/TIE.2014.2370936","article-title":"Electric motor fault detection and diagnosis by kernel density estimation and Kullback-Leibler divergence based on stator current measurements","volume":"62","author":"Giantomassi","year":"2015","journal-title":"IEEE Trans. Ind. Electron."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1360","DOI":"10.1109\/TR.2016.2570549","article-title":"Statistical approach for nondestructive incipient crack detection and characterization using Kullback-Leibler divergence","volume":"65","author":"Harmouche","year":"2016","journal-title":"IEEE Trans. Reliab."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.dsp.2017.06.019","article-title":"Matrix CFAR detectors based on symmetrized Kullback-Leibler and total Kullback-Leibler divergences","volume":"69","author":"Hua","year":"2017","journal-title":"Digit. Signal Process."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.engappai.2015.05.004","article-title":"Electric motor defects diagnosis based on kernel density estimation and Kullback-Leibler divergence in quality control scenario","volume":"44","author":"Ferracuti","year":"2015","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_43","first-page":"231","article-title":"On sparse variational methods and the Kullback-Leibler divergence between stochastic processes","volume":"51","author":"Matthews","year":"2016","journal-title":"J. Mach. Learn. Res."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1109\/18.481781","article-title":"An inequality on guessing and its application to sequential decoding","volume":"42","author":"Arikan","year":"1996","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1016\/S0019-9958(65)90332-3","article-title":"A coding theorem and Renyi\u2019s entropy","volume":"8","author":"Campbell","year":"1965","journal-title":"Inf. Control"},{"key":"ref_46","unstructured":"Renyi, A. (1961). On measures of entropy and information. Proceedings of 4th Berkeley Symposium on Mathematical Statistics and Probability I;, University of California."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"042107","DOI":"10.1103\/PhysRevE.97.042107","article-title":"Relations between heat exchange and R\u00e9nyi divergences","volume":"97","author":"Wei","year":"2018","journal-title":"Phys. Rev. E"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Kumar, M.A., and Sason, I. (2016, January 10\u201315). On projections of the R\u00e9nyi divergence on generalized convex sets. Proceedings of the 2016 IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain.","DOI":"10.1109\/ISIT.2016.7541474"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Sadeghpour, M., Baratpour, S., and Habibirad, A. (2017). Exponentiality test based on Renyi distance between equilibrium distributions. Commun. Stat.-Simul. Comput.","DOI":"10.1080\/03610918.2017.1366514"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"S134","DOI":"10.1016\/S0167-8140(15)30456-4","article-title":"PD-0351: Development of a novel regmentation framework using the Jensen Renyi divergence for adaptive radiotherapy","volume":"111","author":"Markel","year":"2014","journal-title":"Radiother. Oncol."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"610","DOI":"10.1007\/s00145-017-9265-9","article-title":"Improved security proofs in lattice-based cryptography: Using the R\u00e9nyi divergence rather than the statistical distance","volume":"31","author":"Bai","year":"2018","journal-title":"J. Cryptol."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"12472","DOI":"10.1038\/ncomms12472","article-title":"The gravity dual of R\u00e9nyi entropy","volume":"7","author":"Dong","year":"2016","journal-title":"Nat. Commun."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/JHEP01(2018)115","article-title":"Renyi entropy for local quenches in 2D CFT from numerical conformal blocks","volume":"2018","author":"Kusuki","year":"2018","journal-title":"J. High Energy Phys."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"949","DOI":"10.1007\/s00477-016-1221-y","article-title":"One-Dimensional velocity distribution in open channels using Renyi entropy","volume":"31","author":"Kumbhakar","year":"2017","journal-title":"Stoch. Environ. Res. Risk Assess."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.patcog.2016.07.038","article-title":"Selective ensemble of SVDDs with Renyi entropy based diversity measure","volume":"61","author":"Xing","year":"2017","journal-title":"Pattern Recog."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"122","DOI":"10.2174\/2213275910666170307111644","article-title":"An Image Segmentation Method Based on Renyi Relative Entropy and Gaussian Distribution","volume":"10","author":"Nie","year":"2017","journal-title":"Recent Patents Comput. Sci."},{"key":"ref_57","first-page":"277","article-title":"f-entropies, probability of error, and feature selection","volume":"39","year":"1978","journal-title":"Inf. Control"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys"},{"key":"ref_59","first-page":"95","article-title":"Axioms for (\u03b1, \u03b2, \u03b3)-entropy of a generalized probability scheme","volume":"9","author":"Kumar","year":"2013","journal-title":"J. Appl. Math. Stat. Inf."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"505184","DOI":"10.1155\/2014\/505184","article-title":"A generalization of the Havrda-Charvat and Tsallis entropy and its axiomatic characterization","volume":"2014","author":"Kumar","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s00161-004-0174-4","article-title":"Nonextensive statistical mechanics: A brief introduction","volume":"16","author":"Tsallis","year":"2004","journal-title":"Contin. Mech. Thermodyn."},{"key":"ref_62","unstructured":"Rajesh, G., and Sunoj, S.M. (2016). Some properties of cumulative Tsallis entropy of order \u03b1. Stat. Pap."},{"key":"ref_63","doi-asserted-by":"crossref","unstructured":"Singh, V.P. (2016). Introduction to Tsallis Entropy Theory in Water Engineering, CRC Press.","DOI":"10.1201\/b19113"},{"key":"ref_64","unstructured":"Tsonis, A. (2018). Nonextensive Statistical Mechanics: Overview of Theory and Applications in Seismogenesis, Climate, and Space Plasma. Advances in Nonlinear Geosciences, Springer."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"1368","DOI":"10.1016\/j.physa.2017.09.020","article-title":"Text mining by Tsallis entropy","volume":"490","author":"Jamaati","year":"2018","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_66","doi-asserted-by":"crossref","unstructured":"Basu, A., Shioya, H., and Park, C. (2011). Statistical Inference: The Minimum Distance Approach, Chapman & Hall\/CRC.","DOI":"10.1201\/b10956"},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"4394","DOI":"10.1109\/TIT.2006.881731","article-title":"On divergence and information in statistics and information theory","volume":"52","author":"Leise","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_68","unstructured":"Pardo, L. (2006). Statistical Inference Based on Divergences, CRC\/Chapman-Hall."},{"key":"ref_69","unstructured":"Vajda, I. (1989). Theory of Statistical Inference and Information, Kluwer."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1080\/02331880902986919","article-title":"On divergences of finite measures and their applicability in statistics and information theory","volume":"44","author":"Stummer","year":"2010","journal-title":"Statistics"},{"key":"ref_71","unstructured":"Sundaresan, R. (July, January 30). A measure of discrimination and its geometric properties. Proceedings of the IEEE International Symposium on Information Theory, Lausanne, Switzerland."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1109\/TIT.2004.840871","article-title":"Cramear-Rao and moment-entropy inequalities for Renyi entropy and generalized Fisher information","volume":"51","author":"Lutwak","year":"2005","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"5081","DOI":"10.1109\/TIT.2015.2449312","article-title":"Minimization Problems Based on Relative \u03b1-Entropy II: Reverse Projection","volume":"61","author":"Kumar","year":"2015","journal-title":"IEEE Trans. Infor. Theory"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1093\/biomet\/88.3.865","article-title":"A comparison of related density-based minimum divergence estimators","volume":"88","author":"Jones","year":"2001","journal-title":"Biometrika"},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1111\/j.2517-6161.1995.tb02050.x","article-title":"Robustifying model fitting","volume":"57","author":"Windham","year":"1995","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_76","first-page":"1587","article-title":"Normalized estimating equation for robust parameter estimation","volume":"7","author":"Fujisawa","year":"2013","journal-title":"Elect. J. Stat."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"2053","DOI":"10.1016\/j.jmva.2008.02.004","article-title":"Robust parameter estimation with a small bias against heavy contamination","volume":"99","author":"Fujisawa","year":"2008","journal-title":"J. Multivar. Anal."},{"key":"ref_78","unstructured":"Maji, A., Ghosh, A., and Basu, A. (arXiv, 2014). The Logarithmic Super Divergence and its use in Statistical Inference, arXiv."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/s10182-015-0252-x","article-title":"The Logarithmic Super Divergence and Asymptotic Inference Properties","volume":"100","author":"Maji","year":"2016","journal-title":"AStA Adv. Stat. Anal."},{"key":"ref_80","first-page":"39","article-title":"Statistical Inference Based on the Logarithmic Power Divergence","volume":"2","author":"Maji","year":"2017","journal-title":"Rashi"},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"051402","DOI":"10.1103\/PhysRevA.67.051402","article-title":"Anomalous diffusion and Tsallis statistics in an optical lattice","volume":"67","author":"Lutz","year":"2003","journal-title":"Phys. Rev. A"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"110601","DOI":"10.1103\/PhysRevLett.96.110601","article-title":"Tunable Tsallis Distributions in Dissipative Optical Lattices","volume":"96","author":"Douglas","year":"2006","journal-title":"Phys. Rev. Lett."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/j.physa.2005.06.065","article-title":"Triangle for the entropic index q of non-extensive statistical mechanics observed by Voyager 1 in the distant heliosphere","volume":"356","author":"Burlaga","year":"2005","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"055003","DOI":"10.1103\/PhysRevLett.100.055003","article-title":"Superdiffusion and Non-Gaussian Statistics in a Driven-Dissipative 2D Dusty Plasma","volume":"100","author":"Liu","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"097202","DOI":"10.1103\/PhysRevLett.102.097202","article-title":"Generalized Spin-Glass Relaxation","volume":"102","author":"Pickup","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"063001","DOI":"10.1103\/PhysRevLett.102.063001","article-title":"Power-Law Distributions for a Trapped Ion Interacting with a Classical Buffer Gas","volume":"102","author":"Devoe","year":"2009","journal-title":"Phys. Rev. Lett."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"022002","DOI":"10.1103\/PhysRevLett.105.022002","article-title":"Transverse-Momentum and Pseudorapidity Distributions of Charged Hadrons in pp Collisions at \n          \n            \n              \n                \n\t\t\t\t  s\n\t\t\t\t\n              \n            \n          \n         = 7 TeV","volume":"105","author":"Khachatryan","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1007\/JHEP08(2011)086","article-title":"Charged particle transverse momentum spectra in pp collisions at \n          \n            \n              \n                \n\t\t\t\t  s\n\t\t\t\t\n              \n            \n          \n         = 0.9 and 7 TeV","volume":"2011","author":"Chatrchyan","year":"2011","journal-title":"J. High Energy Phys."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"052004","DOI":"10.1103\/PhysRevD.83.052004","article-title":"Measurement of neutral mesons in p + p collisions at \n          \n            \n              \n                \n\t\t\t\t  s\n\t\t\t\t\n              \n            \n          \n         = 200 GeV and scaling properties of hadron production","volume":"83","author":"Adare","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.physletb.2017.10.043","article-title":"Non-extensive statistical mechanics and black hole entropy from quantum geometry","volume":"775","author":"Majhi","year":"2017","journal-title":"Phys. Lett. B"},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1109\/TIT.1980.1056144","article-title":"Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy","volume":"26","author":"Shore","year":"1980","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1063\/1.2423258","article-title":"Updating Probabilities","volume":"872","author":"Caticha","year":"2006","journal-title":"AIP Conf. Proc."},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"180604","DOI":"10.1103\/PhysRevLett.111.180604","article-title":"Nonadditive Entropies Yield Probability Distributions with Biases not Warranted by the Data","volume":"111","author":"Presse","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"052149","DOI":"10.1103\/PhysRevE.90.052149","article-title":"Nonadditive entropy maximization is inconsistent with Bayesian updating","volume":"90","author":"Presse","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"5043","DOI":"10.3390\/e17075043","article-title":"Reply to C. Tsallis\u2019 \u201cConceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems\u201d","volume":"17","author":"Presse","year":"2015","journal-title":"Entropy"},{"key":"ref_96","doi-asserted-by":"crossref","unstructured":"Vanslette, K. (2017). Entropic Updating of Probabilities and Density Matrices. Entropy, 19.","DOI":"10.3390\/e19120664"},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1111\/j.2517-6161.1984.tb01318.x","article-title":"Multinomial goodness-of-fit tests","volume":"46","author":"Cressie","year":"1984","journal-title":"J. R. Stat. Soc. B"},{"key":"ref_98","first-page":"85","article-title":"Eine informations theoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen Ketten","volume":"3","year":"1963","journal-title":"Publ. Math. Inst. Hung. Acad. Sci."},{"key":"ref_99","first-page":"299","article-title":"Information-type measures of difference of probability distributions and indirect observations","volume":"2","year":"1967","journal-title":"Stud. Scientiarum Math. Hung."},{"key":"ref_100","first-page":"329","article-title":"On topological properties of f-divergences","volume":"2","year":"1967","journal-title":"Stud. Scientiarum Math. Hung."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/BF02018661","article-title":"A class of measures of informativity of observation channels","volume":"2","year":"1972","journal-title":"Priodica Math. Hung."},{"key":"ref_102","first-page":"2032","article-title":"Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems","volume":"19","year":"1991","journal-title":"Ann. Stat."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"1081","DOI":"10.1214\/aos\/1176325512","article-title":"Efficiency versus robustness: The case for minimum Hellinger distance and related methods","volume":"22","author":"Lindsay","year":"1994","journal-title":"Ann. Stat."},{"key":"ref_104","first-page":"337","article-title":"A summary of entropy statistics","volume":"31","author":"Esteban","year":"1995","journal-title":"Kybernetica"},{"key":"ref_105","unstructured":"Itakura, F., and Saito, S. (1968, January 21\u201328). Analysis synthesis telephony based on the maximum likelihood method. Proceedings of the 6th International Congress on Acoustics, Tokyo, Japan."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"793","DOI":"10.1162\/neco.2008.04-08-771","article-title":"Nonnegative Matrix Factorization with the Itakura\u2013Saito Divergence: With application to music analysis","volume":"21","author":"Fevotte","year":"2009","journal-title":"Neural Comput."},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1109\/18.179378","article-title":"Convergence of best \u03d5-entropy estimates","volume":"39","author":"Teboulle","year":"1993","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1007\/BF00773476","article-title":"Minimum disparity estimation for continuous models: Efficiency, distributions and robustness","volume":"46","author":"Basu","year":"1994","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.jmva.2008.03.011","article-title":"Parametric estimation and tests through divergences and the duality technique","volume":"100","author":"Broniatowski","year":"2009","journal-title":"J. Multivar. Anal."},{"key":"ref_110","first-page":"600","article-title":"Several applications of divergence criteria in continuous families","volume":"48","author":"Broniatowski","year":"2012","journal-title":"Kybernetika"},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.spl.2008.04.011","article-title":"Optimal robust M-estimators using divergences","volume":"79","author":"Toma","year":"2009","journal-title":"Stat. Probab. Lett."},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/s11634-010-0061-8","article-title":"Optimal robust estimates using the Hellinger distance","volume":"4","author":"Marazzi","year":"2010","journal-title":"Adv. Data Anal. Classif."},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/j.jmva.2012.10.003","article-title":"Optimal robust M-estimators using Renyi pseudodistances","volume":"115","author":"Toma","year":"2010","journal-title":"J. Multivar. Anal."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/5\/347\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:03:31Z","timestamp":1760195011000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/5\/347"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,5,6]]},"references-count":113,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2018,5]]}},"alternative-id":["e20050347"],"URL":"https:\/\/doi.org\/10.3390\/e20050347","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,5,6]]}}}