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Entropy of Degree s and Degree (r,s)

For operational purposes, it seems more natural to consider, the slar expression $\sum_{i=1}^n{p^r_i}$ as an information measure instead of Rényi's entropy of order $r$. So, Havrda and Charvát (1967) [46] proposed the following entropy of degree $s$:

$$H^s(P)=(2^{1-s}-1)^{-1}\bigg[\sum_{i=1}^n{p^s_i}-1\bigg],\ s\neq1,\ s>0,$$ (3.3)

for all $P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$. In these case also, we can easily verify that $\lim_{s\to 1}{H^s(P)}=H(P)$.

This quantity permits a simpler characterization (ref. Havrda and Charvát, 1967) [46]. Daróczy (1970) [33] gave an alternative approach to characterize it (see section 3.5.1).

Sharma and Taneja (1975; 1977) [92], [93] studied a generalization of $H^s(P)$ involving two scalar parameters, known entropy of degree $(r,s)$, and is given by

$$H^{r,s}(P)={(2^{1-r}-2^{1-s})}^{-1}\sum_{i=1}^n{(p^r_i-p^s_i)},\r\neq s,\ r>0,\ s>0$$ (3.4)

for all $P=(p_1,p_2,...,p_n)\ \in\ \Delta_n$, where $r$ and $s$ are real parameters.

In particular, when $r=1$ or $s=1$, the measure (3.4) reduces to (3.3). In the limiting case, we have 

$$\lim_{r\tos}{H^{r,s}(P)}=-2^{r-1}\ \sum_{i=1}^n{p^r_i\ \log{p_i}}, \ r>0,$$ It reduces to Shannon's entropy for $r=1$.