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Unified (r,s)-Relative Information


Rényi (1961) [82] first presented a scalar parametric generalization of Kullback-Leibler's relative information as

$\displaystyle D^1_r(P\vert\vert Q)=(r-1)^{-1}\log\Big(\sum_{i=1}^n{p^r_iq^{1-r}_i}\Big),\r\neq 1,\ r>0,$
for all $ P,Q\ \in\ \Delta_n$, with 
$\displaystyle \lim_{r\to1}{D^1_r(P\vert\vert Q)}=D(P\vert\vert Q).$

Rathie and Kannappan (1972) [79] presented an alternative way to generalize Kullback-Leibler's information as

$\displaystyle D^s_s(P\vert\vert Q)=(1-2^{1-s})^{-1}\Big[\sum_{i=1}^n{p^s_iq^{1-s}_i}-1\Big],\s\neq 1,\ s>0,$
for all $ P,Q\ \in\ \Delta_n$, with 
$\displaystyle \lim_{s\to1}{D^s_s(P\vert\vert Q)}=D(P\vert\vert Q).$

Sharma and Mittal (1977) [91] studied one and two scalar parametric generalizations of $ D(P\vert\vert Q)$ as

$\displaystyle D^s_1(P\vert\vert Q)=(1-2^{1-s})^{-1}\big\{\exp_2\big[(s-1)D(P\vert\vert Q)\big]-1\big\},\s\neq 1,$
and
$\displaystyle D^s_r(P\vert\vert Q)=(1-2^{1-s})^{-1}\Big[\Big(\sum_{i=1}^n{p^r_iq^{1-r}_i}\Big)^{s-1\overr-1}-1\Big],\ r\neq 1,\ s\neq 1,\ r>0,$
for all $ P,Q\ \in\ \Delta_n$, with 
$\displaystyle \lim_{s\to 1}{D^s_r(P\vert\vert Q)}=D^1_r(P\vert\vert Q),$
$\displaystyle \lim_{r\to 1}{D^s_r(P\vert\vert Q)}=D^s_1(P\vert\vert Q),$
$\displaystyle \lim_{s\to1}{D^s_1(P\vert\vert Q)}=D(P\vert\vert Q),$
and 
$\displaystyle D^s_r(P\vert\vert Q)=D^s_s(P\vert\vert Q),\ \ $   for$\displaystyle \ r=s.$

Instead studying the measures $ D(P\vert\vert Q)$,$ D^1_r(P\vert\vert Q)$$ D^s_s(P\vert\vert Q)$$ D^s_1(P\vert\vert Q)$ and $ D^s_r(P\vert\vert Q)$ separately, one can study them jointly. Let us write these measures in the unified expression as in case of entropy-type in the following way:

$\displaystyle {\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)=\left\......rt\vert Q), & r\neq1,\ s=1 \\ D(P\vert\vert Q), & r=1,\ s=1\end{array}\right.$
    (4.1)

for all $ P,\ Q\in\Delta_n,\ r\in(0,\infty)$ and $ s \in (-\infty,\infty)$.

The measure $ D^s_s(P\vert\vert Q)$ does not appear in the unified expression because it is already contained in$ D^s_r(P\vert\vert Q)$ as a particular case when $ r=s$.

We call the measure $ {\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)$, the unified $ (r,s)-$relative information.

Note 4.1. The definition of $ D^s_1(P\vert\vert Q)$ and $ D^s_r(P\vert\vert Q)$ given initially by Sharma and Mittal (1977) [91] involve $ s>0$. But, here in our study we have relaxed this condition. The constant initially considered was$ (2^{s-1}-1)^{-1}\ (s\neq 1)$. Also, we have changed the multiplicative constant and took it as $ (1-2^{1-s})^{-1}\ (s\neq1)$ in order to simplify our study on generalized divergence measures. Some study on the measures $ D(P\vert\vert Q)$$ D^1_r(P\vert\vert Q)$ and$ D^s_s(P\vert\vert Q)$ can be seen in Mathai and Rathie (1975). [71]

In particular, we have

$\displaystyle {\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert U)= n^{s-......mbol{\mathscr{H}}}}^s_r(U)-{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P)\big],$
    (4.2)

for all $ P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$$ U = ({1\overn},...,{1\over n}) \in \Delta_n$, where $ {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r$ is the unified $ (r,s)-$entropy given in Chapter 3, expression (3.8).

As in case of generalized entropies, here also we can write

$\displaystyle {\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)={\ensu......cr{N}}}}_s\big({\ensuremath{\boldsymbol{\mathscr{D}}}}^1_r(P\vert\vert Q)\big),$
    (4.3)

where $ {\ensuremath{\boldsymbol{\mathscr{N}}}}_s:[0,\infty)\ \rightarrow I\!\!R$ (reals) is given by

$\displaystyle {\ensuremath{\boldsymbol{\mathscr{N}}}}_s(x)=\left\{\begin{array}......(1-2^{1-s})^{-1}\big[2^{(s-1)x}-1\big], & s\neq 1\\ x, & s=1\end{array}\right.$
    (4.4)

Proposition 4.1. The measure $ {\ensuremath{\boldsymbol{\mathscr{N}}}}_s(x)$ given by (4.4) has the following properties.

(i) $ {\ensuremath{\boldsymbol{\mathscr{N}}}}_s(x)\geq 0$ with equality iff $ x=0$;
(ii) $ {\ensuremath{\boldsymbol{\mathscr{N}}}}_s(x)$ is an increasing function of x;
(iii) $ {\ensuremath{\boldsymbol{\mathscr{N}}}}_s(x)$ is an increasing function of s;
(iv) $ {\ensuremath{\boldsymbol{\mathscr{N}}}}_s(x)$ is strictly convex function of x for $ s>1$;
(v) $ {\ensuremath{\boldsymbol{\mathscr{N}}}}_s(x)$ is strictly concave function of x for $ s<1$.

21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil