Abstract

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.

摘要

在许多应用中，通常无法获得各个组件的实际故障时间。相反，由于技术和/或经济原因，实际用户仅收集故障总时间数据。在处理此类数据进行可靠性评估时，从业人员经常面临选择潜在故障时间分布和相应的统计推断方法的挑战。到目前为止，由于指数分布，正态分布，伽马分布和逆高斯分布都具有闭合形式的表达式，因此仅用于分析总失效时间数据。但是，概率分布的有限选择不能满足各种工程应用中的广泛需求。在对故障时间数据进行建模时，PHase类型（PH）分布强大而灵活，因为它们可以通过调整模型结构来模拟任意接近的非负随机变量的大量概率分布。在本文中，PH分布首次用于基于总故障时间数据的可靠性估计。开发了最大似然估计（MLE）方法和贝叶斯替代方法。对于MLE方法，开发了一个期望最大化算法进行参数估计，并使用相应的Fisher信息来构造感兴趣量的置信区间。对于贝叶斯方法，还介绍了执行点和间隔估计的过程。