Stress–Strength Reliability Estimation for Gamma Distributions in a Noisy System

Abstract

When assessing the reliability of a system or the viability of a material, it is necessary to consider the stress conditions of the operating system. Let X be the strength of the system and Y be the stress of the system caused by the environment. In a stress–strength model, the system’s reliability is defined as [Formula: see text], and due to ignoring environmental shocks or human failure, it is usually overestimated. To underestimate reliability, we assume that when X goes to the environmental conditions, the system’s strength reduces to [Formula: see text], where Z is a random variable on [Formula: see text]. We propose a modified reliability model with [Formula: see text] for the gamma distribution. In this model, stress Y and the scale mixture of the strength of the gamma distribution, [Formula: see text], are modeled as independent random variables. The proposed model is compared with the traditional model through a simulation study. Additionally, the asymptotic distribution is used to construct an asymptotic confidence interval for [Formula: see text]. Finally, we use two real datasets to demonstrate the practical applicability of our proposed method.