Please use this identifier to cite or link to this item:
|Title:||The First Passage Time of Degradation Processes|
|Department:||Mathematics and Statistics|
|Abstract:||The thesis gives perspectives for the first passage time (FPT) of degradation processes from both the parametric and nonparametric aspects. As an important reliability index of the manufactured products, 100pth percentile of the FPT distribution is always required to provide by the market. If the assumed underlying process is misspecified and fails to fit the degradation data, the estimate of the reliability index will lose efficiency. The typically used degradation processes include the Wiener process, the gamma process, and the inverse Gaussian process because all of them are closed under convolution. This property can help us to obtain the FPT in analytic forms. However, it is difficult to accurately fit the actual data with limited model selections, then it is important to discuss how to achieve the flexible selections of the underlying model. To reduce the misspecification effects for the FPT density, the most straightforward way is to observe the failure times directly. For a good summary and set of references, see Prentice and Kalbfleisch (1979) and Kalbfleisch and Prentice (2002). However, to guarantee the sample size of failure time observation is large enough to make the statistical inference, a large number of experiments are required which are too costly. That is the reason why people choose to reduce the experimental cost by alternatively observing the increments then the FPT density can be estimated based on the mathematical properties of the selected underlying process. Hence, the current methods cannot simultaneously realize the low experimental cost and model robustness. In this thesis, we firstly propose a novel parametric approach which can generalize the degradation processes only if the corresponding Laplace transforms exist. Then the Laplace transform of the FPT density function can be obtained in closed-form and the survival probability can be computed through Laplace inversion. For many stochastic processes, their likelihood functions are intractable so the maximum likelihood estimate (MLE) of the parameters are unavailable to obtain. We estimate the parameters by generalized method of moments (GMM) which is a distance-based method. Specifically, the weighted convolution of two independent gamma processes incorporated with random effects is exemplified as the parametric underlying model, and it is motivated by the scenario of multiple sensors used for monitoring the degradation of the same critical component. Although the degradation processes generated from these sensors reflect the same degradation path, the corresponding scales and noise are significantly different. To find the unified degradation path, Hua et al. (2013) used the weight-averaged unified approach which determined the weights by the defined leadership scores. Then we develop the parametric model into a distribution-free model which can eliminate the misspecification effects caused by wrong process-type assumption. The theoretical Laplace transform of degradation process can be replaced by the empirical Laplace transform composed by the observed increments and the 100pth percentile of the FPT distribution can be approximated by the empirical saddlepoint method. As one of the important applications in reliability engineering, the optimal design for degradation test is studied under both the parametric and nonparametric scenarios. To optimize the degradation test subject to the experimental cost not exceed the pre-specified budget, the design factors in the experiment such as the number of test units, the number of measurements, the inspection frequency and the termination time are considered.|
|Appears in Collections:||Open Access Dissertations and Theses|
Files in This Item:
|Qin_Chengwei_201709_PhD.pdf||818.85 kB||Adobe PDF||View/Open|
Items in MacSphere are protected by copyright, with all rights reserved, unless otherwise indicated.