Estimating a Parametric Component Lifetime Distribution from a Collection of Superimposed Renewal Processes

ABSTRACT

Maintenance data can be used to make inferences about the lifetime distribution of system components. Typically, a fleet contains multiple systems. Within each system, there is a set of nominally identical replaceable components of particular interest (e.g., 2 automobile headlights, 8 dual in-line memory module (DIMM) modules in a computing server, 16 cylinders in a locomotive engine). For each component replacement event, there is system-level information that a component was replaced, but no information on which particular component was replaced. Thus, the observed data are a collection of superpositions of renewal processes (SRP), one for each system in the fleet. This article proposes a procedure for estimating the component lifetime distribution using the aggregated event data from a fleet of systems. We show how to compute the likelihood function for the collection of SRPs and provide suggestions for efficient computations. We compare performance of this incomplete-data maximum likelihood (ML) estimator with the complete-data ML estimator and study the performance of confidence interval methods for estimating quantiles of the lifetime distribution of the component. Supplementary materials for this article are available online.

KEYWORDS:

• Component reliability
• Log-location-scale family
• Maximum likelihood estimation
• Recurrence data
• Relative efficiency
• Superposition of renewal processes

Supplementary Materials

The supplementary materials present simulation studies to evaluate the performance of the idML estimator by studying its efficiency relative to the cdML estimator and the coverage probability performance of Wald approximation and the likelihood ratio test confidence interval methods. The supplementary materials also present the results of another simulation study to investigate the partition probabilities, providing insights to better understand the behavior of the idML estimator.

Acknowledgments

The authors thank the referees, associate editor, editor Peihua Qiu, Dan Nordman, and Vivekananda Roy for providing comments that helped us make important improvements to the article. C codes from Robin K. S. Hankin’s R package “Partitions” were helpful for the implementation idML estimation procedure.