Analysis of systems with dependent components through a variance-based index and regression importance signature

This paper extends a recent regression importance index in reliability theory to subgroups of components, introducing the regression importance signature. This measure identifies, for any fixed number $k$ of components, the subgroup of size $k$ whose joint state most significantly influences system lifetime variability. In addition to this extension, we investigate new properties of the index that enhance understanding of how component reliability and structural role jointly affect system importance. Unlike traditional single-component indices, the measure captures heterogeneity and interactions, reducing them to intuitive behaviours in series or parallel systems and revealing subsystems or dependencies that cannot be detected by individual measures. General results for dependent components are provided, with conditions for comparing individual components and subgroups.