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Entropy of Order r and Degree s

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

Let $ H^s_n:\delta_n \rightarrow I\!\!R$ be a real valued continuous function satisfying

$\displaystyle H(p_1,p_2,...,p_n)=g^{-1}\Big( {\sum_{i=1}^n{p_i\ g(H(\{p_i\}))}\over \sum_{i=1}^n{p_i}} \Big),$

where g is a strictly monotonic continuous function, and $ H(\{p\})\ (0<p\leq 1)$ is the self-information of an event of a probability distribution $ P$ satisfying:

(i) $ H(\{p\})$ is a continuous function $ p$ in (0,1].
(ii) $ H(\{ pq\} )=H(\{ p \})+H(\{ q \})+\lambda H(\{ p\} )H(\{ q\} ), \ \lambda\neq 0.$
(iii) $ H(\{{1\over 2}\})=1.$
$\displaystyle H(p_1,p_2,...,p_n;1,s)=(2^{1-s}-1)^{-1}\Big[ \exp_2\Big((s-1){\sum_{i=1}^n{p_i\log_2 p_i}\over\sum_{i=1}^n{p_i}}\Big)-1\Big],\ s\neq 1,\ s>0$


$\displaystyle H(p_1,p_2,...,p_n;r,s)=(2^{1-s}-1)^{-1}\Big[\Big({\sum_{i=1}^n{p......r \sum_{i=1}^n{p_i}}\Big)^{s-1\overr-1}-1\Big],\ r\neq 1,\ s\neq 1,\ r>0,\ s>0$

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

Let $ H_n:\delta_n \rightarrow I\!\!R$ be a real valued continuous function satisfying the following:

(i) $ H_n(P)$ is a symmetric function of its arguments.
(ii) $ H_1(\{p\})$ is continuous in (0,1].
(iii) $ H_1(\{{1\over 2}\})=1.$
(iv) There is a sequence $ \{f_n\}$ such that
$\displaystyle H_{nm}(P*Q)=H(P)+f_n(p_1,p_2,...,p_n)H_m(Q),$
for all $ P \in \delta_n$$ Q \in \delta_m$ and$ P*Q \in \delta_{nm}$.
(v) There exists a strictly monotonic continuous function g such that



$\displaystyle g(H_n(P))={\sum_{i=1}^n{p_ig\big(H(\{p_i\})\big)}\over\sum_{i=1}^n{p_i}}.$

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 $ H:\Delta_n \rightarrow I\!\!R$ be a real valued continuous function satisfying the following axioms:

(i) $ H(P)=\eta\Big(\sum_{i=1}^n{f(p_i)}\Big)^\delta +\zeta,\ \eta,\delta,\zeta\neq 0,$ where f is a continuous function defined on [0,1].
(ii) $ H(P*Q)=H(P)+H(Q)-{1\over \zeta}H(P)H(Q),$ for $ P \in \ \Delta_n$$ Q\ \in\ \Delta_m$ and $ P*Q\in\Delta_{nm}$.
(iii) $ H({1\over 2},{1\over 2})=1.$
$\displaystyle H(P)=( 2^{(1-\sigma )^\delta}-1)^{-1}\Big[\Big(\sum_{i=1}^n{p^\sigma_i}\Big)^\delta -1\Big],\\sigma>0, \ \delta\neq 0,$
that is same as (3.21) for $ \sigma=r$ and $ \delta={s-1\over r-1}$, with $ \sum_{i=1}^n{p_i}=1$.
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil