## Communications in Mathematical Analysis

### A Generalization of $f$-Divergence Measure to Convex Functions Defined on Linear Spaces

S. S. Dragomir

#### Abstract

In this paper we generalize the concept of $f$-divergence to a convex function defined on a convex cone in a linear space. Some fundamental results are established. Applications for some well known divergence measures are provided as well.

#### Article information

Source
Commun. Math. Anal., Volume 15, Number 2 (2013), 1-14.

Dates
First available in Project Euclid: 9 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.cma/1376053387

Mathematical Reviews number (MathSciNet)
MR3093577

Zentralblatt MATH identifier
1277.26034

Subjects
Primary: 26D15, 94A99

#### Citation

Dragomir, S. S. A Generalization of $f$-Divergence Measure to Convex Functions Defined on Linear Spaces. Commun. Math. Anal. 15 (2013), no. 2, 1--14. https://projecteuclid.org/euclid.cma/1376053387

#### References

• R. Beran, Minimum Hellinger distance estimates for parametric models, Ann. Statist., 5 (1977), 445-463.
• A. Bhattacharyya, On a measure of divergence between two statistical populations defined by their probability distributions, Bull. Calcutta Math. Soc., 35 (1943), 99-109.
• I. Csiszár, 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.
• I. Csiszár, Information-type measures of difference of probability functions and indirect observations, Studia Sci. Math. Hungar, 2 (1967), 299-318.
• I. Csiszár and J. Körner, Information Theory: Coding Theorem for Discrete Memory-less Systems, Academic Press, New York, 1981.
• D. Dacunha-Castelle, Ecole d'été de Probabilités de Saint-Fleour, III-1997, Berlin, Heidelberg: Springer 1978.
• S. S. Dragomir, Semi-inner Products and Applications, Nova Science Publishers Inc., NY, 2004.
• S. S. Dragomir, A new refinement of Jensen's inequality in linear spaces with applications, Mathematical and Computer Modelling 52 (2010), 1497-1505. Preprint RGMIA Res. Rep. Coll. 12(2009), Supplement, Article 6. [Online http://www.staff.vu.edu.au/RGMIA/v12(E).asp].
• S. S. Dragomir, Inequalities for superadditive functionals with applications, Bull. Austral. Math Soc. 77(2008), 401-411.
• S. S. Dragomir, J. Pečarić and L. E. Persson, Properties of some functionals related to Jensen's inequality, Acta Math. Hungarica, 70 (1996), 129-143.
• H. Jeffreys, An invariant form for the prior probability in estimation problems, Proc. Roy. Soc. London, Ser. A, 186 (1946), 453-461.
• J. N. Kapur, A comparative assessment of various measures of directed divergence, Advances in Management Studies, 3 (1984), No. 1, 1-16.
• S. Kullback, Information Theory and Statistics, J. Wiley, New York, 1959.
• S. Kullback and R. A. Leibler, On information and sufficiency, Annals Math. Statist., 22 (1951), 79-86.
• F. Liese and I. Vajda, Convex Statistical Distances, Teubner Verlag, Leipzig, 198
• M. S. Moslehian and M. Kian, Non-commutative $f$-divergence functional, Math. Nachr. (to appear), DOI: 10.1002/mana.201200194.
• A. Renyi, On measures of entropy and information, Proc. Fourth Berkeley Symp. Math. Statist. Prob., Vol. 1, University of California Press, Berkeley, 1961.
• F. Topsoe, Some inequalities for information divergence and related measures of discrimination, Res. Rep. Coll. RGMIA, 2 (1) (1999), 85-98.7.
• I. Vajda, Theory of Statistical Inference and Information, Kluwer, Boston, 1989.