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Unified (r,s)-Inaccuracies

Similar to expression (4.1), we shall give below the three different forms of writing directly the unified$(r,s)-$inaccuracy measures in three different forms:

$$^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)......1 \\ H(P\vert\vert Q)=-\sum_{i=1}^n{p_i\log q_i},& r=1,\ s=1\end{array}\right.$$ (4.5)

$\alpha=1,2$ and 3, where

$$^1M_r(P\vert\vert Q)=\sum_{i=1}^n{p_iq^{r-1}_i},$$

$$^2M_r(P\vert\vert Q)=\Big(\sum_{i=1}^n{p_iq^{(r-1)/r}_i}\Big)^r$$

and

$$^3M_r(P\vert\vert Q)={\sum_{i=1}^n{p^r_i}\over\sum_{i=1}^n{p^r_iq^{1-r}_i}}.$$

The expressions $^1H^1_r(P\vert\vert Q)$, $^2H^1_r(P\vert\vert Q)$ and$^3H^1_r(P\vert\vert Q)$ are due to Nath (1968) [73], Van der Lubbe (1978) [115] and Nath (1975) [74] respectively. The expressions $^1H^s_r(P\vert\vert Q)$ and $H^s_1(P\vert\vert Q)$ are due to Sharma and Gupta (1976) [89].

We can write 

$$^\alpha {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q...... {\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(P\vert\vert Q)\big),\ \alpha=1,2\$$   and$$\ 3$$

where ${\ensuremath{\boldsymbol{\mathscr{N}}}}_s$ is as given in (4.4).

From the unified espression (4.5) it is clear that the measures $^\alpha {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)\ (\alpha=1,2\$   and$\3)$ are continuous with respect to the parameters $r$ and $s$. This allows us to write 

$$^\alpha {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q......symbol{\mathscr{H}}}}^s_r(P\vert\vert Q):r\neq 1,\ s\neq 1\Big\},\ \alpha=1,2\$$   and$$\ 3,$$

where "CE" stands for "continuous extension".

In particular, when $P=Q$, we have 

$$^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)=\,\,......athscr{H}}}}^s_r(P\vert\vert Q)={\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P),$$

and when$Q=P^r$, we have  $$^2{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)={\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P),$$ where $$P^r=\Big( {p^r_1\over \sum_{i=1}^n{p^r_i}},...,{p^r_n\over\sum_{i=1}^n{p^r_i}}\Big) \in \Delta_n.$$