Authors: Raj Mondal, Shivesh Mishra, Aman Chopra
Abstract: Extreme-value theory and reliability analysis are two closely related areas that play a crucial role in modern engineering, environmental science, and risk assessment. While reliability analysis focuses on the probability of survival or failure of systems over time, extreme-value analysis concentrates on modeling rare and extreme events such as maximum loads, peak stresses, floods, or extreme temperatures. These rare events, though infrequent, often dominate system design and safety decisions. This review paper presents a comprehensive overview of the fundamental concepts, statistical models, and applications of extreme-value theory and reliability analysis. Classical distributions used in extreme-value modeling, including Gumbel, Fréchet, and Weibull distributions, are discussed along with their practical relevance. Reliability measures such as reliability function, hazard rate, and mean time to failure are reviewed in detail. The paper also highlights methods for parameter estimation, system reliability modeling, and the integration of extreme-value models into reliability frameworks. Applications in civil engineering, mechanical systems, environmental risk, and industrial safety are presented. The discussion emphasizes both strengths and limitations of existing approaches and outlines future research directions. The presentation is intended to be accessible to researchers and graduate students, though some parts may appear slightly informal in expression.
Keywords: Extreme-value theory, reliability analysis, Weibull distribution, hazard function, risk assessment
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