Analysis of time-censored aggregate data

For complex systems with many components, it is a heavy burden to accurately record all the failure times of each component. To ease the information storage, practitioners in industry may only record the number of failures during an operation interval instead. Such a type of data compresses the detailed failure information and may be called time-censored aggregate data. Because only limited information is available, the statistical inference for aggregate data is more challenging than traditional lifetime data. This study focuses on the statistical analysis of time-censored aggregate data, and four popular parametric lifetime distributions are used to model such data. We first use the gamma distribution and the inverse Gaussian (IG) distribution to model the data, and exploit the maximum likelihood (ML) method for parameter estimation. Then, we fit the data by the Weibull distribution and the log-normal distribution. Unlike the gamma and IG distributions, the Weibull and the log-normal distributions involve multi-dimensional integration in their likelihood functions, which lead to difficulties in estimation. To address the estimation problem, an approximate Bayesian computation algorithm that does not require the likelihood function is proposed. The methods are justified through simulations and validated by a real-life dataset.