Estimating a parametric lifetime distribution from superimposed renewal process data
Maintenance data provides information about the reliability of systems and components. For reparable systems, a failed component will be replaced with a new one, which, in some cases, can be assumed to have the same lifetime distribution as the old component. Estimation of the lifetime distribution is particularly complicated if there are multiple copies of a component installed in multiple locations (different systems and different slots within a system) and information on exactly which component was replaced is not available. In such applications, the replacement history for a collection of locations forms a superposition of renewal processes (SRP). In this dissertation, statistical models and methods motivated by real applications with SRP data were developed for estimating component failure-time distribution. In Chapter 2, the SRP data were modeled by a nonhomogeneous Poisson process (NHPP). This method is motivated by the data structure that can be formulated as a single SRP. The NHPP approximation to the SRP is more adequate as the number of slots in the SRP gets larger. When the SRP has a large number of slots, the ML estimator based on NHPP assumption performs well and the interval estimation procedure based on transformed parametric bootstrap-t method has coverage probabilities close to the nominal values. By comparing the NHPP estimator with an alternative estimator, we make recommendations about which estimator to use for analyzing SRP data. We hope the recommendation provides some guideline for statisticians and engineers for better reliability analysis. In Chapter 3, the exact likelihood for the SRP data is derived. This likelihood-based method is motivated by the data structure that can be formulated as a fleet of independent SRP's. All SRP's in the fleet are assumed to correspond to a same component lifetime distribution. By considering all possible data configurations that could lead to the observed event history, the likelihood can be computed as a weighted sum of conditional likelihoods corresponding to all unique data configurations, with weights being probabilities for each data configuration. We use an ML estimation procedure that starts with a crude estimate of the weights and updates them iteratively. In Chapter 4, we describe an R function that implements the maximum likelihood estimation procedure described in Chapter 3. The need to enumerate all possible data configurations makes the estimation procedure complicated and computationally intensive. We developed an R function that implements the estimation procedure. The function takes the recurrence event data as the input and returns the ML estimates and the estimated covariance matrix of the parameter estimates. Details about the function implementation and how to use the function are discussed.