I'm reading two papers "Quantile based entropy function" and "Quantile based entropy of order statistics". I'm a bit confused whether the quantile based entropy function (Eq 7 in the first paper and Eq 4 in the second) conforms with the standard Shannon differential entropy or not.
The quantile function: $Q(u) = \inf {x : F_X[x] \geq u}$, where $F_X$ is the CDF of $X$. Let $q(u) = dQ(u)/du$ be the quantile density function. The quantile Shannon entropy is: $\int^1_0 \log(q(p)) dp$.
It holds that $q(u) \cdot f(Q(u)) = 1$ for all $u\in [0,1]$.
I'd be happy if anyone could please let me know if the two are distinct objects or the same with an explanation.