  Following the theory of information measures based on the cumulative
distribution function, we propose the fractional generalized cumulative
entropy, and its dynamic version. These entropies are particularly suitable to
deal with distributions satisfying the proportional reversed hazard model. We
study the connection with fractional integrals, and some bounds and comparisons
based on stochastic orderings, that allow to show that the proposed measure is
actually a variability measure. The investigation also involves various notions
of reliability theory, since the considered dynamic measure is a suitable
extension of the mean inactivity time. We also introduce the empirical
generalized fractional cumulative entropy as a non-parametric estimator of the
new measure. It is shown that the empirical measure converges to the proposed
notion almost surely. Then, we address the stability of the empirical measure
and provide an example of application to real data. Finally, a central limit
theorem is established under the exponential distribution.
