Title: Estimation of Different Entropies via Taylor One Point and Taylor Two Points Interpolations Using Jensen Type Functionals
Author(s): Tasadduq Niaz, Khuram Ali Khan, Dilda Pecaric, Josip Pecaric
Pages: 686-710
Cite as:
Tasadduq Niaz, Khuram Ali Khan, Dilda Pecaric, Josip Pecaric, Estimation of Different Entropies via Taylor One Point and Taylor Two Points Interpolations Using Jensen Type Functionals, Int. J. Anal. Appl., 17 (5) (2019), 686-710.

Abstract


In this work, we estimated the different entropies like Shannon entropy, Renyi divergences, Csiszar divergence by using the Jensen’s type functionals. The Zipf’s mandelbrot law and hybrid Zipf’s mandelbrot law are used to estimate the Shannon entropy. Further the Taylor one point and Taylor two points interpolations are used to generalize the new inequalities for m-convex function.

Full Text: PDF

 

References


  1. Anderson, G., & Ge, Y. The size distribution of Chinese cities. Reg. Sci. Urban Econ., 35 (2005), 756-776. Google Scholar

  2. Auerbach, F. (1913). Das Gesetz der Bevlkerungskonzentration. Petermanns Geographische Mitteilungen, 59 (2005), 74-76. Google Scholar

  3. Black, D., & Henderson, V. Urban evolution in the USA. J. Econ. Geogr., 3 (2003), 343-372. Google Scholar

  4. Bosker, M., Brakman, S., Garretsen, H., & Schramm, M. A century of shocks: the evolution of the German city size distribution 19251999. Reg. Sci. Urban Econ., 38 (2008), 330-347. Google Scholar

  5. Butt, S. I., Khan, K. A., & Pecaric, J. Generaliztion of Popoviciu inequality for higher order convex function via Tayor’s polynomial, Acta Univ. Apulensis Math. Inform., 42 (2015), 181-200. Google Scholar

  6. Butt, S. I., Mehmood, N., & Pecaric, J. New generalizations of Popoviciu type inequalities via new green functions and Fink’s identity. Trans. A. Razmadze Math. Inst., 171 (2017), 293-303. Google Scholar

  7. Butt, S. I., & Pecaric, J. Popoviciu’s Inequality For N-convex Functions. Lap Lambert Academic Publishing, (2016). Google Scholar

  8. Butt, S. I., & Pecaric, J. Weighted Popoviciu type inequalities via generalized Montgomery identities. Rad Hazu. Mat. Znan., 19 (2015), 69-89. Google Scholar

  9. Butt, S. I., Khan, K. A., & Pecaric, J. Popoviciu type inequalities via Hermite’s polynomial. Math. Inequal. Appl., 19 (2016), 1309-1318. Google Scholar

  10. Horvath, L. A method to refine the discrete Jensen’s inequality for convex and mid-convex functions. Math. Computer Model., 54 (2011), 2451-2459. Google Scholar

  11. Horvath, L., Khan, K. A., & Pecaric, J. Combinatorial Improvements of Jensens Inequality / Classical and New Refinements of Jensens Inequality with Applications, Monographs in inequalities 8, Element, Zagreb. (2014). Google Scholar

  12. Horvath, L., Khan, K. A., & Pecaric, J. Refinement of Jensen’s inequality for operator convex functions. Adv. Inequal. Appl., 2014 (2014), Art. ID 26. Google Scholar

  13. Horvath, L., Pecaric, J. A refinement of discrete Jensen’s inequality, Math. Inequal. Appl. 14 (2011), 777-791. Google Scholar

  14. Ioannides, Y. M., & Overman, H. G. Zipf’s law for cities: an empirical examination. Reg. Sci. Urban Econ., 33 (2003), 127-137. Google Scholar

  15. Csiszar, I. Information measures: a critical survey. In: Tans. 7th Prague Conf. on Info. Th., Statist. Decis. Funct., Rand. Proc. 8th Eur. Meeting Stat., Vol. B (1978), 73-86. Google Scholar

  16. Csiszar. I. . Information-type measures of difference of probability distributions and indirect observations. Stud. Sci. Math. Hungar. 2 (1967), 299-318. Google Scholar

  17. Horvath, L., Pecaric, D. & Pecaric, J. Estimations of f-and Renyi divergences by using a cyclic refinement of the Jensen’s inequality. Bull. Malaysian Math. Sci. Soc., 42 (2019). 933-946 . Google Scholar

  18. Khan, K. A., Niaz, T., Pecaric, ¯D., Pecaric, J. Refinement of Jensen’s Inequality and Estimation of f- and Renyi Divergence via Montgomery identity. J. Inequal. Appl., 2018 (2018), Art. ID 318. Google Scholar

  19. Kullback, S. Information theory and statistics. Courier Corporation. Google Scholar

  20. Kullback, S., & Leibler, R. A. (1951). On information and sufficiency. Anna. Math. Stat., 22 (1997), 79-86. Math. Dokl. 4(1963), 121-124. Google Scholar

  21. Lovricevic, N., Pecaric, D. & Pecaric, J. ZipfMandelbrot law, f-divergences and the Jensen-type interpolating inequalities. J. Inequal. Appl., 2018 (2018), Art. ID 36. Google Scholar

  22. Matic, M., Pearce, C. E., & Pecaric, J. Shannon’s and related inequalities in information theory. In Survey on Classical Inequalities (pp. 127-164). Springer, Dordrecht. (2000). Google Scholar

  23. Niaz, T., Khan, K. A., & Pecaric, J. On generalization of refinement of Jensen’s inequality using Fink’s identity and Abel-Gontscharoff Green function. J. Inequal. Appl., 2017 (2017), Art. ID 254. Google Scholar

  24. Pecaric, J., Proschan, F., & Tong, Y. L. Convex functions, Partial Orderings and Statistical Applications, Academic Press, New York. (1992). Google Scholar

  25. Renyi, A. On measure of information and entropy. In: Proceeding of the Fourth Berkely Symposium on Mathematics, Statistics and Probability, pp. 547-561. (1960). Google Scholar

  26. Rosen, K. T., & Resnick, M. The size distribution of cities: an examination of the Pareto law and primacy. J. Urban Econ., 8 (1980), 165-186. Google Scholar

  27. Soo, K. T. Zipf’s Law for cities: a cross-country investigation. Reg. Sci. Urban Econ., 35 (2005), 239-263. Google Scholar

  28. Zipf, G. K. Human behaviour and the principle of least-effort. Cambridge MA edn. Reading: Addison-Wesley. (1949). Google Scholar