Nair and Sathar (2020) introduced a new metric for uncertainty known as dynamic failure extropy, focusing on the analysis of past lifetimes. In this study, we extend this concept to a bivariate context, exploring various properties associated with the proposed bivariate measure. We show that bivariate conditional failure extropy can uniquely determine the joint distribution function. Additionally, we derive characterizations for certain bivariate lifetime models using this measure. A new stochastic ordering, based on bivariate conditional failure extropy, is also proposed, along with some established bounds. We further develop an estimator for the bivariate conditional failure extropy using a smoothed kernel and empirical approach. The performance of the proposed estimator is evaluated through simulation studies.

翻译：Nair和Sathar（2020）引入了一种称为动态失效外熵的不确定性新度量，侧重于对过去寿命的分析。在本研究中，我们将此概念扩展到二元情境，探索与所提出的二元度量相关的各种性质。我们证明二元条件失效外熵能够唯一确定联合分布函数。此外，我们利用该度量推导了某些二元寿命模型的表征。我们还提出了一种基于二元条件失效外熵的新随机序，并建立了一些界限。我们进一步使用平滑核与经验方法，为二元条件失效外熵开发了一个估计量。通过模拟研究评估了所提出估计量的性能。