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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)  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) .

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) . The measures ( =1,2 and 3) are the three different one parametric generalizations of the Kerridge's (1961)  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)  and Sharma and Gupta (1976)  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) .

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