Quantile-based Fractional Generalized Cumulative Past Entropy

Abstract: Uncertainty in past lifetime distributions and the timing of inactivity in systems and their components can be effectively measured using the fractional generalized cumulative past entropy (FGCPE) and its dynamic extension (DFGCPE), introduced by Di Crescenzo et al. (2021). Building on this framework, we propose a quantile-based variant, the quantile fractional generalized cumulative past entropy (QFGCPE), along with its dynamic time-dependent counterpart (DQFGCPE). Closed-form expressions of these measures are derived analytically for a variety of lifetime distributions, including those with and without explicit distribution functions. Fundamental properties such as bounds, monotonicity, and stochastic orderings are investigated to assess robustness and interpretability. Furthermore, we construct a nonparametric estimator of QFGCPE and establish its asymptotic validity through extensive simulation studies involving bias, mean squared error (MSE), and root mean squared error (RMSE). Finally, the sensitivity of the proposed QFGCPE measure is examined by comparing its behavior with the logistic map, demonstrating its ability to capture transitions from order to chaos.