Date of Graduation


Document Type


Degree Name

Doctor of Philosophy in Engineering (PhD)

Degree Level



Industrial Engineering


Haitao Liao

Committee Member

Edward A. Pohl

Second Committee Member

Avishek Chakraborty

Third Committee Member

Xiao Liu


Aggregate failure-time data, Bayesian method, Maximum likelihood estimation, Phase-type distribution


In particular engineering applications, such as reliability engineering, complex types of data are encountered which require novel methods of statistical analysis. Handling covariates properly while managing the missing values is a challenging task. These type of issues happen frequently in reliability data analysis. Specifically, accelerated life testing (ALT) data are usually conducted by exposing test units of a product to severer-than-normal conditions to expedite the failure process. The resulting lifetime and/or censoring data are often modeled by a probability distribution along with a life-stress relationship. However, if the probability distribution and life-stress relationship selected cannot adequately describe the underlying failure process, the resulting reliability prediction will be misleading. To seek new mathematical and statistical tools to facilitate the modeling of such data, a critical question to be asked is: Can we find a family of versatile probability distributions along with a general life-stress relationship to model complex lifetime data with covariates? In this dissertation, a more general method is proposed for modeling lifetime data with covariates. Reliability estimation based on complete failure-time data or failure-time data with certain types of censoring has been extensively studied in statistics and engineering. However, 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 challenges of selecting the underlying failure-time distributions and the corresponding statistical inference methods.

So far, only the Exponential, Normal, Gamma and Inverse Gaussian (IG) distributions have been used in analyzing aggregate failure-time data because these distributions have 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 nonnegative random variables arbitrarily closely by adjusting the model structures. In this paper, PH distributions are utilized, for the first time, in reliability estimation based on aggregate failure-time data. To this end, a maximum likelihood estimation (MLE) method and a Bayesian alternative are developed. For the MLE method, an expectation-maximization (EM) algorithm is developed to estimate the model parameters, 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. Several 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.