Analysis of strength distributions of multi-modal failures using the EM algorithm

Abstract

Analysis of various multi-modal strength distributions are studied by using competing risks models. This multi-modality may arise due to several kinds of flaws in a material. The fracture of a material is controlled by the most severe of all the flaws, the so-called ‘weakest-link theory’, which is also commonly referred to as ‘competing risks’ in the statistics literature. These multi-modal problems can also be further complicated due to possible censoring. In practice, censoring is very common because of time and cost considerations on experiments. Moreover, in certain situations, it is observed that the mode of failure is not properly identified due to lack of appropriate diagnostics, expensive and time-consuming autopsy, etc. This is known as the masking problem. Several studies have been carried out, but they have mainly focused on bi-modal Weibull distributions with no censoring or masking considered. In this paper, we deal with the strength distribution of multi-modal failures when censoring and masking are present. We provide the EM-type parameter estimator for a variety of strength distributions including Weibull, lognormal and inverse Gaussian distributions. The applicability of this method is illustrated by several examples.

Keywords:

• Competing risks
• Censoring
• Masking
• EM algorithm
• MLE
• Missing data
• Likelihood function
• Weibull
• Lognormal
• Inverse Gaussian (Wald)

Acknowledgements

W. J. Padgett's work was partially supported by the National Science Foundation grant DMS-0243594 to the University of South Carolina. The authors thank Michael Harwell of Chemical Engineering at Clemson University for providing the bond strength data set.