Inspection-conditioned uncertainty and complexity in coherent systems under double periodic inspections: extropy-based measures

Abstract Reliability assessment under scheduled inspections is fundamentally an inference problem, because remaining lifetime must be evaluated conditionally on partial evidence revealed at inspection times. This paper develops a signature-based information-theoretic framework for coherent systems observed under two periodic inspections and studies the conditional residual lifetime beyond the second inspection. Uncertainty is quantified using cumulative residual extropy, yielding explicit representations and sharp comparison results that link inspection timing and system architecture to uncertainty reduction. To capture structural heterogeneity beyond uncertainty magnitude, we introduce divergence measures built from cumulative residual extropy that quantify complexity induced by coherent design under a common inspection policy. Stochastic-ordering results provide rigorous ranking tools for both inspection-conditioned lifetimes and the induced mixing structure. For statistical inference, we propose nonparametric plug-in and kernel-smoothed estimators, prove their large-sample behavior, and assess finite-sample accuracy by Monte Carlo experiments. An application to medical image intensities shows that the same two-inspection protocol produces distinct uncertainty and complexity profiles that discriminate between clinically relevant photometric transformations.