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The actual failure times of individual components are usually unavailable in many applications. Instead, only aggregate failure-time data are collected by actual users, due to technical and/or economic reasons. When dealing with such data for reliability estimation, practitioners often face the challenges of selecting the underlying failure-time distributions and the corresponding statistical inference methods. So far, only the exponential, normal, gamma and inverse Gaussian distributions have been used in analyzing aggregate failure-time data, due to these distributions having closed-form expressions for such data. However, the limited choices of probability distributions cannot satisfy extensive needs in a variety of engineering applications. PHase-type (PH) distributions are robust and flexible in modeling failure-time data, as they can mimic a large collection of probability distributions of non-negative random variables arbitrarily closely by adjusting the model structures. In this article, PH distributions are utilized, for the first time, in reliability estimation based on aggregate failure-time data. A Maximum Likelihood Estimation (MLE) method and a Bayesian alternative are developed. For the MLE method, an Expectation-Maximization algorithm is developed for parameter estimation, and the corresponding Fisher information is used to construct the confidence intervals for the quantities of interest. For the Bayesian method, a procedure for performing point and interval estimation is also introduced. Numerical examples show that the proposed PH-based reliability estimation methods are quite flexible and alleviate the burden of selecting a probability distribution when the underlying failure-time distribution is general or even unknown. 

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. 

The length-of-stay (LOS) is an important quality metric in health care, and the use of phase-type (PH) distribution provides a flexible method for modeling LOS. In this paper, we model the patient flow information collected in a hospital for patients of distinct diseases, including headache, liveborn infant, alcohol abuse, acute upper respiratory infection, and secondary cataract. Based on the results obtained from fitting Coxian PH distributions to the LOS data, the patients can be divided into different groups. By analyzing each group to find out their common characteristics, the corresponding readmission rate and other useful information can be evaluated. Furthermore, a comparison of patterns for each disease is analyzed. We conclude that it is important to offering better service and avoiding waste of sources, by the analysis of the relations between groups and readmission. In addition, comparing the patterns within distinct diseases, a better decision for assigning resources and improving the insurance policy can be made.