Next:Unified InaccuraciesUp:Generalized Information Measures

Properties of Unified (r,s)-Relative Information

The unified relative information given by (4.1) satisfies the following properties;

Property 4.1. (Continuity). is a continuous function of the pair and is also continuous with respect to the parameters and .

Property 4.2. (Symmetry). is a symmetric function of their arguments in the pair , i.e., where is an arbitrary permutation of to .

Property 4.3. (Expansibility). We can write  for all  and  .

Property 4.5. (Nonnegativity). with equality iff .

Property 4.6. (Monotonicity). is an increasing function of ( fixed) and of ( fixed). In particular, when , the result still holds.

Property 4.7. (Inequalities among the measures). We have

(i) (ii) Property 4.8. (Convexity). is a convex function of the pair of probability distributions for .

Property 4.9. (Generalized data processing inequality). We have where and are the probability distributions given by and where    is a stochastic matrix such that  .

Property 4.10. (Schur-convexity) is a Schur-convex function in the pair .

Property 4.11. For , we have Property 4.12. (Order preserving) We have

(i) If , then for all , where and are determined by the equations: and (ii) If , then where 21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil