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Monotonicity of the Jensen functional for $$f$$-divergences with applications to the Zipf-Mandelbrot law. (English) Zbl 1430.94045
Summary: The Jensen functional in its discrete form is brought in relation to the Csiszár divergence functional via its monotonicity property. Thus deduced general results branch into specific forms for some of the well known $$f$$-divergences, e.g. the Kullback-Leibler divergence, the Hellinger distance, the Bhattacharyya coefficient, $$\chi^2$$-divergence, total variation distance. Obtained comparative inequalities are also interpreted in the environment of the Zipf and the Zipf-Mandelbrot law.
##### MSC:
 94A15 Information theory (general) 94A17 Measures of information, entropy 26D15 Inequalities for sums, series and integrals 26A51 Convexity of real functions in one variable, generalizations
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##### References:
 [1] I. CSISZAR´,Information-type measures of difference of probability functions and indirect observations, Studia Sci. Math. Hungar.,2(1967), 299-318. · Zbl 0157.25802 [2] I. CSISZAR´,Information measures: A critical survey, Trans. 7th Prague Conf. on Info. Th. Statist. Decis. Funct., Random Processes and 8th European Meeting of Statist., Volume B, Academia Prague, 1978, 73-86. [3] S. S. DRAGOMIR,Some inequalities for the Csisz´arΦ−divergence, RGMIA. [4] L. EGGHE, R. ROUSSEAU,Introduction to Informetrics. Quantitative Methods in Library, Documentation and Information Scince, Elsevier Science Publishers, New York, (1990). [5] T.VANERVEN, P. HARREMOES¨,R´enyi Divergence and Kullback-Leibler Divergence, Journal of Latex Class Files,61 (2007); arXiv: 1206.2459v2 [cs.IT]24 April 2014. [6] S. KULLBACK,Information Theory and Statistics, J.Wiley, New York, 1959. · Zbl 0088.10406 [7] S. KULLBACK, R. A. LEIBLER,On information and sufficiency, Annals of Mathematical Statistics, 221 (1951), 79-86. · Zbl 0042.38403 [8] N. LOVRICEVIˇC´, -D. PECARIˇC´, J. PECARIˇC´,Zipf-Mandelbrot law, f−divergences and the Jensentype interpolating inequalities, J. Inequal. Appl. (2018) 2018:36 https://doi.org/10.1186/s13660-0181625-y [9] B. MANARIS, D. VAUGHAN, C. S. WAGNER, J. ROMERO, R. B. DAVIS,Evolutionary Music and the Zipf-Mandelbrot Law: Developing Fitness Functions for Pleasant Music, Proceedings of 1st European Workshop on Evolutionary Music and Art (EvoMUSART2003), 522-534. · Zbl 1033.68858 [10] B. MANDELBROT,An informational theory of the statistical structure of language, In Jackson, W., editor, Communication Theory, New York, Academic Press, 1953., 486-502. [11] B. MANDELBROT,Information Theory and Psycholinguistics, in Wolman, B.B., and Nagel, E. (eds.), Scientific psychology, Basic Books, 1965. [12] D. YU. MANIN,Mandelbrot’s Model for Zipf’s Law: Can Mandelbrot’s Model Explain Zipf’s Law for Language?, Journal of Quantitative Linguistics,163 (2009), 274-285. [13] D. S. MITRINOVIC´, J. E. PECARIˇC´, A. M. FINK,Classical and New Inequalities in Analysis, Kluwer Academic Publishers, Dordrecht/Boston/London, 1993. [14] M. A. MONTEMURRO,Beyond the Zipf-Mandelbrot law in quantitative linguistics, URL:arxiv:condmat/0104066v2, (2001). · Zbl 0978.68126 [15] D. MOUILLOT, A. LEPRETRE,Introduction of relative abundance distribution (RAD) indices, estimated from the rank-frequency diagrams(RFD), to assess changes in community diversity, Environmental Monitoring and Assessment, Springer63(2) (2000), 279-295.
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