Gumbel mixture modelling for multiple failure data

Abstract

The Gumbel mixture model is a popular tool for modelling extreme events with mixed formation mechanisms. The lack of a shape parameter in the Gumbel distribution makes it somewhat less practical for mixture modelling compared to the Weibull distribution, which is widely used in structural reliability and safety applications. Since the Gumbel distribution has two forms (i.e., left- and right-skewed), we describe it as a three-parameter distribution. The additional parameter ξ∈{−1,1} is used to control the skewness of the Gumbel distribution. The maximum likelihood parameter estimation procedure for the proposed Gumbel mixture model is derived. Based on the simulation study, we confirmed that 1) the estimation procedure can successfully estimate the parameters of the mixture model, and 2) the proposed three-parameter Gumbel distribution has advantages over the state of the art in mixture modelling using different parametric families. An illustrative real-world example is also considered. Climbing rope failure was observed as a function of two random variables, and two competing failure modes were found. The experiment and subsequent mixture modelling provided further insight into the degradation of climbing ropes. Finally, the proposal is implemented in the freely available R package rebmix.