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M-Dimensional Unified (r,s)-Arithmetic and Geometric Mean Divergences


The three different ways of defining unified$ (r,s)-A$$ G-$divergences or unified $ (r,s)-T-$divergence measures are given by

$\displaystyle ^\alpha{\ensuremath{\boldsymbol{\mathscr{T}}}}^s_r(P_1,P_2,...,P_......P_M), & r\neq 1,\ \ s=1 \\ T(P_1,P_2,...,P_M), &r=1,\ \ s=1\end{array}\right.$
    (5.23)

$ \alpha=1,2$ and 3, where for $ \alpha =1$, we have

$ ^1T^s_r(P_1,P_2,...,P_M)$

$\displaystyle =(1-2^{1-s})^{-1}\Big\{\Big[ \sum_{i=1}^n{ \Big(\sum_{j=1}^M{......lambda_kp^{1-r}_{ki}}\Big)\Big]^{{s-1}\over{r-1}}}-1\Big\},\r\neq 1,\ s\neq 1$
$\displaystyle ^1T^s_1(P_1,P_2,...,P_M)=(1-2^{1-s})^{-1}\Big\{2^{(s-1)T(P_1,P_2,...,P_M)}-1\Big\},\, s\neq 1$
and
$\displaystyle ^1T^1_r(P_1,P_2,...,P_M)=(r-1)^{-1}\log_2\Big\{\sum_{i=1}^n{\Big(......_{ji}}\Big)^r\Big(\sum_{k=1}^M{\lambda_kp^{1-r}_{ki}}\Big)}\Big\}, \, r\neq 1$
For $ \alpha=2$, we have
$\displaystyle ^2{\ensuremath{\boldsymbol{\mathscr{T}}}}^s_r(P_1,P_2,...,P_M)=\s......dsymbol{\mathscr{D}}}}^s_r \Big(\sum_{j=1}^M{\lambda_jP_j}\vert\vert P_k \Big).$
    (5.24)

For $ \alpha =3$, we have

$ ^3 T^s_r(P_1,P_2,...,P_M)$

$\displaystyle =(1-2^{1-s})^{-1}\Big\{\Big[\sum_{i=1}^n{\Big(\sum_{j=1}^M{\l......lambda_k}}\Big)^{1-r}\Big]^{{s-1}\over{r-1}}}-1\Big\}, \, r\neq 1, \, s\neq 1$
$\displaystyle ^3T^s_1(P_1,P_2,...,P_M)=(1-2^{1-s})^{-1}\Big\{2^{(s-1)T(P_1,P_2,...,P_M)}-1\Big\},\, s\neq 1$
and
$\displaystyle ^3T^1_r(P_1,P_2,...,P_M)=(r-1)^{-1}\log_2\Big\{\sum_{i=1}^n{\Big(......}\Big)^r\Big(\prod_{k=1}^M{p_{ki}^{\lambda_k}}\Big)^{1-r}} \Big\}, \, r\neq 1$
In particular, we have 

$\displaystyle ^1 {\ensuremath{\boldsymbol{\mathscr{T}}}}^s_r(P_1,P_2,...,P_M)=\......(P_1,P_2,...,P_M)= {\ensuremath{\boldsymbol{\mathscr{T}}}}^s_s(P_1,P_2,...,P_M)$

where
$\displaystyle {\ensuremath{\boldsymbol{\mathscr{T}}}}^s_s(P_1,P_2,...,P_M)=\lef......1,P_2,...,P_M), & \ s\neq 1 \\  T(P_1,P_2,...,P_M), & s=1\end{array}\right.$
with 
$\displaystyle T^s_s(P_1,P_2,...,P_M)=(1-2^{1-s})^{-1}\Big\{\sum_{i=1}^n{ \Big......ij}}\Big)^s\Big(\sum_{k=1}^M{\lambda_k p^{1-s}_{ki}}\Big)}-1\Big\},\ s\neq1$
For two dimensional case, i.e., when M=2,$ \lambda_1=\lambda_2={1\over 2},\ P_1 =P$ and $ P_2=Q$, we have

$ ^1 T^s_r(P_1,P_2)=\ ^1T^s_r(P\vert\vert Q)$

$\displaystyle = (1-2^{1-s})^{-1}\Big\{ \Big[ \sum_{i=1}^n {\Big( {{p_i+q_i}\ove......g)^r\Big( {{p^{1-r}_i+q^{1-r}_i}\over 2}\Big)}\Big]^{{s-1}\over{r-1}}-1\Big\},$
    (5.25)

$ ^2 T^s_r(P_1,P_2)=\ ^2T^s_r(P\vert\vert Q)$

$\displaystyle = (1-2^{1-s})^{-1}{1\over 2}\Big\{ \Big[ \sum_{i=1}^n {\Big({{p_......^n {\Big( {{p_i+q_i}\over 2}\Big)^r p^{1-r}_i}\Big]^{{s-1}\over{r-1}}-2 \Big\},$
    (5.26)
and

$ ^3 T^s_r(P_1,P_2)=\ ^3T^s_r(P\vert\vert Q)$

$\displaystyle = (1-2^{1-s})^{-1}\Big\{ \Big[ \sum_{i=1}^n{ \Big( {{p_i+q_i}\over2} \Big)^r\Big(\sqrt{p_iq_i}\Big)^{1-r}}\Big]^{{s-1}\over{r-1}}-1\Big\},$
    (5.27)
for all $ r\neq 1, \ s\neq 1$
21-06-2001
Inder Jeet Taneja
Departamento de Matemática - UFSC
88.040-900 Florianópolis, SC - Brazil