International Journal of Engineering, Science and M athematics
Vol. 7Issue 1,January 2018,
ISSN: 2320-0294 Impact Factor: 6.765
Journal Homepage: http://www.ijesm.co.in , Email: ijesmj@gmail.com Double-
Blind Peer Reviewed Refereed Open Access International Journal - Included in the International Serial Directories Indexed & Listed at:
Ulrich's Periodicals Directory © , U.S.A., Open J-Gage as well as in Cabell’s Directories of Publishing Opportunit ies, U.S.A
441
International Journal of Engineering, Science and Mathematics
http://www.ijesm.co.in , Email: ijesmj@gmail.com
Generalized 'useful’ AG and 'useful' JS-Divergence M easures and their
Bounds
Reetu Kumari
D.K. Sharma
Abstract
In the present paper we consider one parametric generalization of
'useful' AG and 'useful' JS-divergence measures. Both the
generalizations can be expressed as particular cases of Csiszar f-
divergence. Under some conditions of probability distributions,
some bounds inequalities are also obtained.
M athematics Subject Classification (2000): 94A17 and 26 D15
Keywords:
Csiszar’s f-divergence;
Useful’ Relative information;
KL Divergence;
Inequalities;
Convex function.
Copyright © 2018 International Journals of Multidisciplinary
Research Academy. All rights reserved.
Author correspondence:
D.K. Sharma,
Jaypee University of Engineering and Technology
A.B. Road, Raghogarh, Dist. Guna – M .P. (India) – 473226,
1. Introduction
Let
1 2
1
( , ,..., ), 0, 1,
n
n n i i
i
p p p p p p
be a set of all possible discrete probability distributions of a
random variable X having utility distribution
1 2
( , ,..., ); 0 for all
n i
U u u u u i attached to each
n
P
such that 0
i
u is utility of an event having probability of occurrence 0
i
p .
The following measure of ‘useful’ directed divergence or ‘useful’ relative information is given by:
1
1
log
( ; ; )
n
i
i i
i i
n
i i
i
p
up
q
IPQU
up
(1)
It may be noted that (1) is a generalization of the measure
1
1
log
( ; )
n
i i i
i
n
i i
i
up p
HPU
up
(2)
It can be observed that (1) is not symmetric in P and
Q
and its symmetric version is given by
Jaypee University of Engineering and Technology A.B. Road, Raghogarh, Dist. Guna – M.P. (India) – 473226
Jaypee University of Engineering and Technology A.B. Road, Raghogarh, Dist. Guna – M .P. (India) – 473226