Degree Type


Date of Award


Degree Name

Doctor of Philosophy





First Advisor

William Q. Meeker


Maintenance data can be used to make inferences about the reliability of system components. In industrial reliability applications, it is common that a fleet contains multiple systems. Within each system, there are multiple copies of a component installed in multiple locations (sockets). Examples are two automobile headlights, eight DIMM modules in a computing server, sixteen cylinders in a locomotive engine, etc. For each component replacement event, there is system-level information that a component was replaced. The socket-level information (which particular component was replaced), however, is usually unknown. The aggregated data for a system form a collection of superpositions of renewal processes (SRP). The reliability of the system component is of particular interest for future system design and maintenance planning. In this dissertation, statistical models and methods were developed for estimating the lifetime distribution of system components for the SRP data, which is motivated by some real applications.

In Chapter 2, we propose a parametric likelihood-based procedure for estimating the lifetime distribution of a component from the aggregated recurrence data for a fleet with multiple SRPs. We show how to formulate the likelihood function of the SRP data and compute the maximum likelihood (ML) estimator. The performance of the ML estimator is investigated by simulation studies, as well as real applications on two different data sets. Chapter 3 provides more extensive results of simulation studies on the performance of the ML estimator and two confidence interval methods for estimating quantiles of the component lifetime distribution.

Chapter 4 presents two graphical distributional assessment methods to evaluate how well the parametric models proposed in Chapter 2 fit the observed SRP data. The first method provides a flexible semi-parametric estimate of the lifetime distribution of the component, which is based on a piecewise exponential (PEX) model. The nonparametric simultaneous confidence bands (NPSCBs) based on a bootstrap procedure, are given to assess the amount of statistical uncertainty of the semi-parametric estimate. The second method is based on the use of the mean cumulative function (MCF) plot for the SRP data. The amount of sampling error can be evaluated by using NPSCBs, which show the departures from the fitted model.

Chapter 5 describes how the maximum likelihood estimation procedure (proposed in Chapter 2) for the SRP data is implemented in the SRPML R package. Given the required input, which includes the observed SRP data, the specified log-location-scale distribution and certain parameter specifications, the functions in the SRPML package return the parameter ML estimates, the maximized log likelihood, and confidence intervals based on Wald and/or likelihood-based procedures. A real application of the R package is illustrated, and the required computation time for this estimation procedure are summarized.

Copyright Owner

Wei Zhang



File Format


File Size

120 pages