On the study of bivariate and conditional dynamic negative cumulative extropy

Abstract Failure extropy, introduced by Nair and Sathar Nair [(2020). On dynamic failure extropy. J. Indian Soc. Probab. Stat. 21: 287–-313], provides a complementary perspective to entropy for quantifying uncertainty in lifetime distributions. However, it becomes mathematically invalid for distributions with unbounded support. To overcome this limitation, Tahmasebi and Toomaj [(2022). On negative cumulative extropy with applications. Commun. Stat. Theory Methods 51(15): 5025-–5047] proposed the concept of negative cumulative extropy (NCEx), offering a bounded and interpretable alternative. In this paper, we extend the notion of NCEx to the bivariate dynamic setting, where uncertainty is assessed for systems whose components have failed at specified times. The proposed formulation effectively captures the uncertainty associated with past lifetimes under dependence, which the existing NCEx cannot address. The measure is further generalized to a vector-valued form, and its fundamental properties are established, including monotonicity, invariance, bounds expressed in terms of the expected inactivity time, and key characterizations. A new stochastic ordering based on the proposed measure is also established. To facilitate practical implementation, a nonparametric estimator is developed and its performance evaluated through extensive Monte Carlo simulations. The practical relevance of the proposed measure is demonstrated using a real dataset, and its superiority over existing entropy-based approaches is shown on an additional dataset.