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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

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. (Schur-convexity) We have

(i)  is Schur-convex function in  for and is Schur-concave in  for .
(ii)  is Schur-convex in  for .
(iii)  is Schur-convex in  for .
Property 4.22. (Shannon-Gibbs-type 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 Kullback-Leibler'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].

21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil