An equation associated with the distance between probability distributions
Abstract
In this paper, we solve the functional equation
f1(pr,qs) + f2(ps,qr) = g(p,q)h(r,s) (p,q,r,s∈]0,1])
where f1,f2,g,h are complex-valued functions defined on ]0,1]. This functional equation is a generalization of a functional equation which was instrumental in the characterization of symmetric divergence of degree α in [3]. This equation arises in the characterization of symmetric weighted divergence of degree α and symmetric inset divergence of degree α.
References
2. J.K. Chung, PL. Kannappan, C.T. Ng, A generalization of the cosine-sine functional equation on groups, Linear Algebra and Its Applications 66 (1985), 259-277.
3. J.K. Chung, PL. Kannappan, C.T. Ng, P.K. Sahoo, Measures of distance between probability distributions, J. Math. Anal. Appl. 139 (1989), 280-292.
4. H. Jeffreys, An invariant form for the prior probability in estimation problems, Proc. Roy. Soc. London Ser. A 186 (1946), 453-461.
5. PL. Kannappan, P.K. Sahoo, Kullback-Leibler type distance measures between probability distributions, Jour. Math. Phy. Sci. 26 (1992), 443-454.
6. PL. Kannappan, P.K. Sahoo, J.K. Chung, On a functional equation associated with the symmetric divergence measures, Utilitas Math. 44 (1993), 75-83.
7. S. Kullback, Information theory and statistics, Peter Smith, Gloucester, MA, 1978.
8. S. Kullback, R.A. Leibler, On information and sufficiency, Annals. Math. Statist. 22 (1951), 79-86.
Department of Pure Mathematics, University of Waterloo, Canada Canada
Department of Mathematics, University of Louisville, USA United States
Department of Applied Mathematics, South China University of Technology, People's Republic of China China
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
