Parameter estimation from load-sharing system data using the expectation–maximization algorithm

Abstract

This article considers a system of multiple components connected in parallel. As components fail one by one, the remaining working components share the total load applied to the system. This is commonly referred to as load sharing in the reliability engineering literature. This article considers the traditional approach to the modeling of a load-sharing system under the assumption of the existence of underlying hypothetical latent random variables. Using the Expectation–Maximization (EM) algorithm, a methodology is proposed to obtain the maximum likelihood estimates in such a model in the case where the underlying lifetime distribution of the components is lognormal or normal. The proposed EM method is also illustrated and substantiated using numerical examples. The estimates obtained using the EM algorithm are compared with those obtained using the Broyden–Fletcher–Goldfarb–Shanno algorithm, which falls under the class of numerical methods known as Newton or quasi-Newton methods. The results show that the estimates obtained using the proposed EM method always converge to a unique global maximizer, whereas the estimates obtained using the Newton-type method are highly sensitive to the choice of starting values and thus often fail to converge.

Keywords:

• Broyden–Fletcher–Goldfarb–Shanno algorithm
• Expectation–Maximization algorithm
• load sharing
• hypothetical latent random variable
• maximum likelihood estimate
• lognormal distribution