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研究生： 黃偉恆
Wei-heng Huang
論文名稱： 具有韋伯壽命零件的串聯系統之可靠度分析
Reliability Analysis of a Series System with Weibull Lifetime Components
指導教授： 樊采虹
Tsai-Hung Fan
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 統計研究所
Graduate Institute of Statistics
畢業學年度： 97
語文別： 英文
論文頁數： 59
中文關鍵詞： EM 演算法 、有母數拔靴法 、type-I 設限 、貝氏估計 、串聯系統 、韋伯分佈 、最大概似估計量 、可靠度函數。
外文關鍵詞： parametric bootstrap method, EM-algorithm, reliability function., maximum likelihood estimators, Weibull distribution, Series system, Type-I censoring, Bayesian estimators
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    • 在串聯系統中，只要有一個零件失效，系統即無法運作。然而每個零件可能有不同的失效時間分佈，使得系統停擺的時間往往是不確定的。
    本文考慮具有m個零件的串聯系統，假設每個零件之壽命具韋伯分佈且彼此獨立，若只觀察到系統失效之時間時之可靠度分析。
    首先，以EM演算法求得各零件壽命分佈中參數的最大概似估計量及相關統計推論；另外並考慮以無資訊先驗分佈的
    "馬可夫鏈蒙地卡羅"方法之客觀貝氏推論。　
    更進一步地將上述方法，發展於 type-I 設限實驗中，另以有母數的拔靴法以估計該模型下參數最大概似估計量的標準差。
    模擬結果顯示，在兩種實驗中，最大概似推論與貝氏推論都能提供準確的估計，而樣本不是太大時，無資訊先驗分佈之貝氏分析所得結果較最大概似法為佳。

    A series system fails if any of its components fails. However, each component may have different life time distribution and,
    in practice, the exact component responsible for the failure of the system can not often be identified. This paper considers
    a life test on a series system of m components, each having a Weibull life time distribution, and when only the system failure
    time is observed. The maximum likelihood estimates via EM algorithm is developed for the parameters of each component as well as
    for the system reliability. Objective Bayesian inference incorporated with the Markov chain Monte Carlo method is also addressed.
    Furthermore, statistical inference is developed for the Type-I censoring experiment within a prespecified time interval and
    parametric bootstrap method is used to estimate the standard errors of the MLE in this case.
    Simulation study carried out reveals that the Bayesian analysis with noninformative prior provides better results than the likelihood approach
    in both situations at least in the case of small sample sizes.

    摘要i Abstract ii 誌謝iii Contents iv List of Tables vi List of Figures vii 1. Introduction 1 1.1 Statistical Motivation and Background . . . . . . . . . . . . . . . . . . . . . 1 1.2 Proposed Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Complete Scheme 5 2.1 Model Description and Notation . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Complete observations . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 EM algorithm under incomplete observations . . . . . . . . . . . . . . 10 2.2.3 Inference on functions of parameters . . . . . . . . . . . . . . . . . . 12 2.3 Bayesian Approach via Markov Chain Monte Carlo Method . . . . . . . . . . . 13 2.4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.1 Simulation study I: Complete model . . . . . . . . . . . . . . . . . . 15 2.4.2 Simulation study II: Incomplete model . . . . . . . . . . . . . . . . . . 20 2.4.3 Illustrative example . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3. Type-I Censoring Scheme 27 3.1 Model Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 Statistical Inference under Complete Failure Observations . . . . . . . . . . . 29 3.2.1 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . . 29 3.2.2 Parametric bootstrap method . . . . . . . . . . . . . . . . . . . . . . 31 3.2.3 Bayesian approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.4 Inference on functions of parameters . . . . . . . . . . . . . . . . . . 34 3.3 Inference without Knowing the Causing Component Experiment . . . . . . . . 35 3.3.1 EM algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.2 Bayesian approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4.1 Simulation study I: Complete model under Type-I censoring . . . . . . 37 3.4.2 Simulation study II: Incomplete model under Type-I censoring . . . . . 43 4. Conclusion 47 Reference 48

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