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Properties of Unified (r,s)Inaccuracies
The unified inaccuracy
measures
and
satisfy the following properties:
Property 4.13. (Continuity)
and
are continuous functions of the pair
and are also continuous with respect to the parameters
and .
Property 4.14. (Symmetry)
and
are symmetric function of their arguments in the pair ,
i.e.,
and
where
is an arbitrary permutation from
to .
Property 4.15. (Expansibility) For
and ,
we have
Property 4.16. (Nonadditivity) We have
and
for ,
and
Property 4.17. (Nonnegativity)and
with equality iff ,
where
is a distribution such that one of the probabilities is one and all others
are zero.
Property 4.18. (Monotonicity)and )
are monotonically increasing functions of
(
fixed).
Property 4.19. We have

(i)
is a convex function of
for
and is concave function of
for.

(ii)
and
are convex functions of
for .
Property 4.20. (Pseudoconvexity). We have

(i)
is pseudoconvex function in Q for
and is pseudoconcave in Q for .

(ii)
is pseudoconvex in Q for .

(iii)
is pseudoconvex in Q for .
Note 4.2. It is well known that (ref. Mangasarian, 1969) [68]
every pseudoconvex function is quasiconvex and every pseudoconcave function
is quasiconcave. Thus for the respective values of the parameters
and
in the above results the quasiconvexity (resp. quasiconcavity) follows.
Property 4.21. (Schurconvexity) We have

(i)
is Schurconvex function in
for and
is Schurconcave in
for .

(ii)
is Schurconvex in
for .

(iii)
is Schurconvex in
for .
Property 4.22. (ShannonGibbstype inequalities) We have

(i) ,
under the following condition:

(ii)
and 3.
Note 4.3. The part (i) of the above property can also be proved
under the condition,
for
using the techniques similar to Kapur (1987) [54].
Property 4.23. (Inequalities among the measures). For
and ,
we have

(i)

(ii)
where
is given by

(iii)
for
=1 and 2.

(iv) ,
with equality iff either
or .

(v) ,

(vi)

(vii) .

(viii) .

(ix) .
Note 4.4. The measures
and
are the one parametric generalizations of the Shannon's entropy and KullbackLeibler's
directed divergence respectively studied by Rényi (1961) [82].
The measures
(=1,2
and 3) are the three different one parametric generalizations of the Kerridge's
(1961) [63] inaccuracy. The parts (vi) and
(vii) connects these measures in an interesting way. But, unfortunately,
these are not extendable for the unified measures.
Moreover, the part (vii) don't have sense for ,
because in this case, the L.H.S. becomes negative. Also for ,
the inequality (viii) have sense, provided the L.H.S. remains positive.
Note 4.5. Sharma and Mittal (1977) [91]
and Sharma and Gupta (1976) [89] considered
the properties (3.22) and (3.34) along with other postulates to characterize
the measures
and
respectively. For a simplified characterization of
refer to Taneja (1984b) [102].
21062001
Inder Jeet Taneja
Departamento de Matemática  UFSC
88.040900 Florianópolis, SC  Brazil