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## Entropy of Order r and Degree s

Sharma and Mittal (1975) [90] presented an axiomatic characterization of entropy of order  and degree. It is based on the Rényi's approach, where the additivity property has been changed (generally referred as nonadditivity).

Let  be a real valued continuous function satisfying

where g is a strictly monotonic continuous function, and  is the self-information of an event of a probability distribution  satisfying:

(i)  is a continuous function  in (0,1].
(ii)
(iii)
Then
 (3.20)

and

 (3.21)

Van der Pyl (1977) [118]restructured the above axiomatic system and considered as follows:

Let  be a real valued continuous function satisfying the following:

(i)  is a symmetric function of its arguments.
(ii)  is continuous in (0,1].
(iii)
(iv) There is a sequence  such that
for all  and.
(v) There exists a strictly monotonic continuous function g such that

Then the above set of axioms lead to the measures (3.20) and (3.21).

Picard (1979) [77] extended the above set of axioms by introducing the idea of weights or preferences and came up with the weighted entropies given in section 3.6.2.

The measure (3.21) can be characterized in a much more simplified form given as follows:

Let  be a real valued continuous function satisfying the following axioms:

(i)  where f is a continuous function defined on [0,1].
(ii)  for  and .
(iii)
Then
that is same as (3.21) for  and , with .
21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil