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Entropy of Degree s

The following characterization of the measure (3.3) is due to Havrda and Charvát (1967) [46].

A function $ H(p_1,p_2,...,p_n;s)$ will be said structural $ s-$entropy if

(i) $ H(p_1,p_2,...,p_n;s)$ is continuous in the region $ \Delta_n$ for $ s>0$.
(ii) $ H(1,s)=0;\ H({1\over 2},{1\over 2};s)=1.$
(iii) $ H(p_1,p_2,...,p_{i-1},0,p_{i+1},...,p_n;s)$
$\displaystyle =H(p_1,p_2,...,p_{i-1},p_{i+1},...,p_n;s),\ i=1,2,...,n.$
(iv) $ H(p_1,p_2,...,p_{i-1},v_{i_1},v_{i_2},p_{i+1},...,p_n;s)$
$\displaystyle =H(p_1,p_2,...,p_{i-1},p_i,p_{i+1},...,p_n;s)+c\,\,p^s_iH\Big({v_{i_1}\over p_i}, {v_{i_2}\over p_i}\Big),$
$\displaystyle v_{i_1}+v_{i_2}=p_i>0,\ i=1,2,...,n,\ c>0. $
Axioms (i)-(iv) determine the structural $ s-$entropy given by (3.3)

In the Rényi's case, Daróczy (1963) [31] restructured the axioms, while in the Havrda and Charvt's case, Daróczy (1970) [33] presented an alternative way to characterize the entropy of degree $ s$. This is as follows:

Let $ H^s_n:\Delta_n \rightarrow $$ I\!\!R$$ n\geq 2$ be a real valued function satisfying:

$\displaystyle H^s_n(p_1,p_2,...,p_n)=\sum_{t=2}^n{(p_1+p_2+...+p_t)\psi_s\Big({p_i\over p_1+p_2+...p_t}\Big)},$
where $ \psi_s:[0,1)\rightarrow$$ I\!\!R$ is a function such that
$\displaystyle \psi_s(p)+(1-p)^s\,\psi_s({q\over1-p})=\psi_s(q)+(1-q)^s\,\psi_s({p\over 1-q}),\ s>0$
for $ p,q \in[0,1),\ p+q\leq 1,$ with $ \psi_s(0)=\psi_s(1),\ \psi_s({1\over2})=1$, then
$\displaystyle \psi_s(p)=(2^{1-s}-1)^{-1}[p^s+(1-p)^s-1],\ s\neq 1,\ s>0$

and $ H^s_n(P)$ is as given by (3.3).

The function $ \psi_s(p)$ given by (3.16) is famous as "information function of degree $ s$".

An alternative way to characterize the entropy of degree $ s$ is following the Chaundy and McLeod's (1961) [27] approach. This can be seen in Sharma and Taneja (1975) [92], and is as follows:

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

$\displaystyle H^s_{nm}(P*Q)=H^s_n(P)+H^s_m(Q)+(2^{1-s}-1)H^s_n(P)H^s_m(Q),\s\neq 1,\ s>0$

for all $ P \in \Delta_n, Q \in \Delta_m,\ P*Q \in\Delta_{nm}$, and 

$\displaystyle H^s_n(P)=\sum_{i=1}^n{f_s(p_i)},$
where$ f_s:[0,1) \rightarrow I\!\!R$ is real valued continuous function with $ f_s({1\over 2})=1$. Then $ H^s_n(P)$ is given by (3.3).

Some alternative approaches to characterize the measure (3.51) can be seen in Aczél and Daróczy (1975) [2], Mathai and Rathie (1975) [71] and Taneja (1979) [99].

It is worth emphasizing here that the measure (3.3), or better $ {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s(P)$$ P \in \ \Delta_n$ satisfies some extra properties given below.

Property 3.18. We can write 

$\displaystyle {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s(P)=\sum_{i=1}^n{f_s}(p_i),$
$\displaystyle f_s(p)=\left\{\begin{array}{ll}(2^{1-s}-1)^{-1}(p^s-p), & s\neq 1,\ s>0 \\  -p\ \log_2p, & s=1\end{array}\right.$
Property 3.19. We can write 
$\displaystyle {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s(p_1,p_2,...,p_n)={\en......{\mathscr{H}}}}^s_s\Big({p_1\over p_1+p_2},{p_1\over p_1+p_2}\Big),\p_1+p_2>0$

Property 3.20. Let $ \psi_s(p) = {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s(p,1-p)$, then

(i) $ \psi_s(p)=\psi_s(1-p).$

(ii) $ \psi_s(0)=\psi_s(1)=0.$

(iii) $ \psi_s({1\over 2})=1.$

(iv) $ \psi_s(p)+(1-p)^s\,\, \psi_s({q\over 1-p})=\psi_s(q)+(1-q)^s\,\,\psi_s({p\over 1-q}),\ p,q\in[0,1),\ p+q\leq 1.$

(v) $ {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s(p_1,p_2,...,p_n)=\sum_{t=2}^n{(p_1+p_2+...+p_t)^s}\,\,\psi_s\Big({p_i\over p_1+p_2+...+p_t}\Big).$
Property 3.21. Let 
$\displaystyle \phi_s(n)=H^s_s({1\overn},...,{1\over n})=({1\over n})^s \\sum_{t=1}^n\,\,{t^s\psi_s({1\over t})},$
with $ \psi_s(1)=0;\\psi_s(2)=1$. Then 
$\displaystyle \psi_s({m\over n})=\phi_s(n)-({m\over n})^s \\phi_s(m)-(1-{m\over n})^s\,\,\phi_s(n-m),$
for all rationals$ \frac{m}{n} \in(0,1),\,\, (1\leq m<n).$

Property 3.22. For all $ P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$$ (w_{k1},...,w_{km})\ \in \Delta_m$,$ k=1,2,...,n$, we have

(i) $ {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s(p_1w_{11},...,p_1w_{1m},p_2w_{21},...,p_2w_{2m},...,p_nw_{n1},...,p_nw_{nm})$ 
$\displaystyle ={\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s(p_1,p_2,...,p_n)+\sum_{k=1}^n{p^s_k}\,{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s(w_{k1},...,w_{km}).$
(ii) $ {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s\Big(\sum_{k=1}^n{p_kw_{k1}},\ \sum......_{kt}{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s(w_{k1},...,w_{km}) },\ s\geq 1$
Property 3.23. For all $ (v_{11},...,v_{nm})\ \in\\Delta_{nm}$, we have

$ {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s(v_{11},...,v_{1m},v_{21},...,v_{2m},...,v_{n1},...,v_{nm})$

$\displaystyle \leq {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_s\Big(\sum_{j=1}^m......mbol{\mathscr{H}}}}^s_s\Big(\sum_{k=1}^n{v_{k1}},...,\sum_{k=1}^n{v_{kn}}\Big),$
for all $ s\geq 1$.

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