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Properties of Unified (r,s)-Inaccuracies

The unified $(r,s)-$inaccuracy measures$^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$$(\alpha=1,2$ and $\3)$ satisfy the following properties:

Property 4.13. (Continuity)$^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$$(\alpha=1,2$ and $\3)$ are continuous functions of the pair $(P,Q)$ and are also continuous with respect to the parameters $r$ and $s$.

Property 4.14. (Symmetry)$^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$$(\alpha=1,2$ and $\3)$ are symmetric function of their arguments in the pair $(P,Q)$, i.e., 

$$^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(p_1,...,p_n\ve......_{\tau_{(n)}} \vert\vert q_{\tau_{(1)}},...,q_{\tau_{(n)}}),\,\,(\alpha=1,2\,$$and$$\, 3)$$ where $\tau$ is an arbitrary permutation from $1$ to $n$.

Property 4.15. (Expansibility) For$(\alpha=1,2$ and $\3)$, we have 

$$^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(p_1,...,p_n,0\......a{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(p_1,...,p_n\vert\vert q_1,...,q_n)$$

Property 4.16. (Nonadditivity) We have

$^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P_1*P_2\vert\vert Q_1*Q_2) ......_1)\, + \,^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P_2\vert\vert Q_2)$

$$+\, (2^{1-s}-1)\,\, ^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}......a{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P_2\vert\vert Q_2),\ (\alpha=1,2\$$and$$\ 3)$$ for $P_1,\ Q_{1}\in \Delta_n$, $P_2,\Q_{2} \in \Delta_m$ and $P_1*P_2,\ Q_1*Q_{2} \in \Delta_{nm}.$

Property 4.17. (Nonnegativity)$^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q) \geq 0$$(\alpha=1,2 \,\,$and$\,\,3)$ with equality iff $P=Q=P^{0} \in \Delta_n$, where $P^0$ is a distribution such that one of the probabilities is one and all others are zero.

Property 4.18. (Monotonicity)$^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$$(\alpha=1 \$and $2$) are monotonically increasing functions of $r$ ($s$ fixed).

Property 4.19. We have

(i) $^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$ is a convex function of $Q\ \in\ \Delta_n$ for $0<r<2, \ s<2$ and is concave function of $Q\ \in\ \Delta_n$ for$r>2, \ s>2$.

(ii) $^{\alpha}{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$ $(\alpha=2$ and $\3)$ are convex functions of $Q\ \in\ \Delta_n$ for $r>0, \ s<2$.

Property 4.20. (Pseudoconvexity). We have

(i) $^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$ is pseudoconvex function in Q for $0<r<2$ and is pseudoconcave in Q for $r>2$.

(ii) $^2{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$ is pseudoconvex in Q for $r>0$.

(iii) $^3{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$ is pseudoconvex in Q for $r>0$.

Note 4.2. It is well known that (ref. Mangasarian, 1969) [68] every pseudoconvex function is quasiconvex and every pseudoconcave function is quasiconcave. Thus for the respective values of the parameters $r$ and $s$ in the above results the quasiconvexity (resp. quasiconcavity) follows.

Property 4.21. (Schur-convexity) We have

(i) $^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$ is Schur-convex function in $Q$ for $r>1$and is Schur-concave in $Q$ for $0<r<2$.

(ii) $^2{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$ is Schur-convex in $Q$ for $r>0$.

(iii) $^3{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$ is Schur-convex in $Q$ for $r>0$.

Property 4.22. (Shannon-Gibbs-type inequalities) We have

(i) ${\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P) \leq \, ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$, under the following condition: 

$$\sum_{i=1}^n{p^r_i}\geq\sum_{i=1}^n{q^r_i},$$

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

Note 4.3. The part (i) of the above property can also be proved under the condition$\sum_{i=1}^n{{p_i\over q_i}}\geq n$, for $r \in (0,1)$ using the techniques similar to Kapur (1987) [54].

Property 4.23. (Inequalities among the measures). For $\alpha =1$ and $2$, we have

(i) $^\alpha {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)\left\{\begi......ldsymbol{\mathscr{H}}}}^s_1(P\vert\vert Q), &0<r \leq 1\end{array}\right.$

(ii) $^\alpha {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)\left\{\begi......{\boldsymbol{\mathscr{H}}}}^1_r(P\vert\vert Q), & s\geq 1\end{array}\right.$

where $k(s)$ is given by
$k(s) = \left\{\begin{array}{ll} \geq (2^{1-s}-1)^{-1}(1-s)\,ln\,2, & s \neq 1, \\ 1, & s =1\end{array}\right.$

(iii) $^\alpha {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q) \left\{\beg......mathscr{H}}}}^1_r(P\vert\vert Q)\leq 1,\ s\leq 1\big)\\\end{array}\right.$ for $\alpha$ =1 and 2.

(iv) $^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)\leq \ ^2{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$, with equality iff either $r=1$ or $Q=({1\over n},...,{1\over n})\in \Delta_n$.

(v) ${\ensuremath{\boldsymbol{\mathscr{D}}}}^s_r(P\vert\vert Q)\leq \ ^3{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P\vert\vert Q)$,

(vi) $max\big\{ {\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(P),{\ensuremath{\boldsym......^1_r(P\vert\vert Q)+{\ensuremath{\boldsymbol{\mathscr{D}}}}^1_r(P\vert\vert Q).$

(vii) $(r-1){\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(Q)\leq \ ^1{\ensuremath{\bold......}^1_r(P\vert\vert Q)+{\ensuremath{\boldsymbol{\mathscr{D}}}}^1_r(P\vert\vert Q)$.

(viii) ${\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(P)+(r-1){\ensuremath{\boldsymbol{\......}}}}^1_r(Q)\leq r\ ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(P\vert\vert Q)$.

(ix) ${\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(P) \leq r\ ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(P\vert\vert Q),\ r\geq 1$.

Note 4.4. The measures ${\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(P)$ and ${\ensuremath{\boldsymbol{\mathscr{D}}}}^1_r(P\vert\vert Q)$ are the one parametric generalizations of the Shannon's entropy and Kullback-Leibler's directed divergence respectively studied by Rényi (1961) [82]. The measures$^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^1_r(P\vert\vert Q)$ ($\alpha$=1,2 and 3) are the three different one parametric generalizations of the Kerridge's (1961) [63] inaccuracy. The parts (vi) and (vii) connects these measures in an interesting way. But, unfortunately, these are not extendable for the unified $(r,s)-$measures. Moreover, the part (vii) don't have sense for $0<r<1$, because in this case, the L.H.S. becomes negative. Also for $0<r<1$, the inequality (viii) have sense, provided the L.H.S. remains positive.

Note 4.5. Sharma and Mittal (1977) [91] and Sharma and Gupta (1976) [89] considered the properties (3.22) and (3.34) along with other postulates to characterize the measures $D^s_r(P\vert\vert Q)$ and $^1H^s_r(P\vert\vert Q)$ respectively. For a simplified characterization of $D^s_r(P\vert\vert Q)$ refer to Taneja (1984b) [102].