On the study of bivariate and conditional failure extropy with application

In recent years, the complementary dual of entropy, known as extropy, has emerged as a valuable tool for quantifying uncertainty in probability distributions. The present work examines the behaviour of failure extropy for bivariate and multivariate random vectors under dependence structures and derives corresponding theoretical bounds. We also introduce a novel vector-valued bivariate dynamic failure extropy (BDFEx), whose components, termed conditional failure extropy (CFEx), capture component-wise conditional uncertainty. For CFEx, we derive several bounds and characterization results, thereby contributing to its theoretical foundation. A new stochastic order based on CFEx is also introduced and investigated. To support empirical analysis, we propose an estimator for CFEx, establish its consistency, and examine its bias and variance. Its performance is further evaluated through Monte Carlo simulations, demonstrating its accuracy under various scenarios. Furthermore, the practical applicability of the proposed measure is demonstrated using real-world biomedical data, along with a comparative evaluation against an entropy-based approach. In addition, a new dependence measure based on bivariate failure extropy (BFEx) is introduced and compared with classical dependence measures, such as the Pearson correlation coefficient, Kendall's tau, and Spearman's rho, demonstrating its superior effectiveness in specific cases.