Weighted Fractional Generalized Cumulative Past Entropy

In this paper, we introduce weighted fractional generalized cumulative past entropy of a nonnegative absolutely continuous random variable with bounded support. Various properties of the proposed weighted fractional measure are studied. Bounds and stochastic orderings are derived. A connection between the proposed measure and the left-sided Riemann-Liouville fractional integral is established. Further, the proposed measure is studied for the proportional reversed hazard rate models. Next, a nonparametric estimator of the weighted fractional generalized cumulative past entropy is suggested based on the empirical distribution function. Various examples with a real life data set are considered for the illustration purposes. Finally, large sample properties of the proposed empirical estimator are studied.