Past extropy for linear consecutive r-out-of-n: F systems and its properties, bounds, and estimation

This paper develops a rigorous information-theoretic framework for uncertainty quantification in lifetime analysis through the concept of past extropy. Particular attention is paid to linear consecutive $r$-out-of- $n$: $F$ systems with independent and identically distributed component lifetimes. Explicit analytical expressions for the past extropy of system lifetimes are derived, and several new theoretical properties are established, including monotonicity results, probabilistic bounds, and characterization theorems, which reveal structural relationships between system reliability configurations and information-based uncertainty measures. The investigation is further extended to conditional past extropy, providing additional insight into uncertainty assessment under partial system information. To support practical implementation, a nonparametric kernel-based estimator of past extropy is proposed. Its performance is evaluated through Monte Carlo simulation experiments that illustrate stable finite-sample behavior and reliable estimation accuracy across different parameter settings. The presented results demonstrate that past extropy offers a flexible and informative framework for studying uncertainty in lifetime distributions and structured reliability systems, combining rigorous theoretical developments with practical data-driven estimation methodology.