Weighted fractional generalized cumulative past entropy and its properties

TL;DR

This paper introduces a new weighted fractional measure of past entropy for nonnegative continuous variables, explores its properties, and develops an estimator with large sample analysis.

Contribution

It proposes the weighted fractional generalized cumulative past entropy, studies its properties, and provides a nonparametric estimator with theoretical large sample results.

Findings

Derived bounds and stochastic orderings for the measure

Established connection with Riemann-Liouville fractional integral

Demonstrated estimator performance with real data examples

Abstract

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 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.