Enhancing reliability analysis with limited observations: A statistical framework for system safety margins

The reliability analysis of complex systems is crucial for cost-effective evaluations, particularly when using deterministic black-box models. This study examines system performance under uncertainty, where the input vector x∈X⊂R^d defines both system and environmental conditions, and failure is characterized by F=x∈X∣y(x)⩽s^⋆, with y the variable of interest and s^⋆∈R a given threshold. Since x is uncertain, a probabilistic analysis is required to ensure robust safety assessments. Such an analysis typically involves two key steps: first, estimating the system's probability of failure (noted p_f), and then, evaluating it against safety standards or expert knowledge. While considerable effort has been invested in proposing efficient methods for estimating p_f, little attention has been paid to the decision phase, which should take into account the uncertainties. This work focuses on the definition and use of safety margins in system reliability analysis with a final decision making purpose, especially when the knowledge of the input vectors x is limited to a finite set of n observations. A key distinction is made between cases where n is large or small relative to 1/p_f. The main contributions of the paper focus on scenarios with small n and propose two approaches for defining reasonable safety margins. The first estimates the probability distribution of x, while the second, based on extreme value theory, directly assesses the tail behavior of the output distribution. The proposed framework is validated through numerical case studies.