A robust approach for estimating field reliability using aggregate failure time data

Failure time data of fielded systems are usually obtained from the actual users of the systems. Due to various operational preferences and/or technical obstacles, a large proportion of field data are collected as aggregate data instead of the exact failure times of individual units. The challenge of using such data is that the obtained information is more concise but less precise in comparison to using individual failure times. The most significant needs in modeling aggregate failure time data are the selection of an appropriate probability distribution and the development of a statistical inference procedure capable of handling data aggregation. Although some probability distributions, such as the Gamma and Inverse Gaussian distributions, have well-known closed-form expressions for the probability density function for aggregate data, the use of such distributions limits the applications in field reliability estimation. For reliability practitioners, it would be invaluable to use a robust approach to handle aggregate failure time data without being limited to a small number of probability distributions. This paper studies the application of phase-type (PH) distribution as a candidate for modeling aggregate failure time data. An expectation-maximization algorithm is developed to obtain the maximum likelihood estimates of model parameters, and the confidence interval for the reliability estimate is also obtained. The simulation and numerical studies show that the robust approach is quite powerful because of the high capability of PH distribution in mimicking a variety of probability distributions. In the area of reliability engineering, there is limited work on modeling aggregate data for field reliability estimation. The analytical and statistical inference methods described in this work provide a robust tool for analyzing aggregate failure time data for the first time.