Weighted fractional generalized cumulative past entropy and its properties

In this paper, we introduce weighted fractional generalized cumulative past entropy of a nonnegative absolutely continuous random variable. Various properties of the proposed weighted fractional measure are studied, including some bounds and stochastic orders. 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 model. Next, a nonparametric estimator of the weighted fractional generalized cumulative past entropy is proposed based on empirical distribution function. Various examples with a real-life data set are considered for illustrative purpose. A validation of the proposed measure is provided using the logistic map and some applications are also discussed. Weighted fractional generalized cumulative paired entropy is proposed and some of its properties are explored. Finally, large-sample properties of the proposed empirical estimator are studied.