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## Entropy of Order r

The following characterization is due to Rényi (1961) [82]. Let us consider the following postulates defined for the function , where

 (3.15)
(i) H(P) is a symmetric function of the elements of P.
(ii) If  denotes the generalized probability distribution consisting of the single probability  that  is a continuous function of p in the interval .
(iii) .
(iv) For  and, we have
Before stating the last postulate, we introduce some notations. Let , and  be two generalized probability distributions such that , we have
with  where  is strictly monotonic function.

Then

Later Daróczy (1963; 1964) [31], [32] reformed the above axiomatic system. Based on the same motivations of Rényi, later researchers (Aczél and Daróczy, 1963 [1]; Varma,1966 [119]; Kapur, 1967 [54] and Rathie, 1970 [78]) generalized the entropy of order  having more than one parameter. These generalizations are specified in the list of entropies (see section 3.5.2).

2001-06-21