壽命具二元Marshall-Olkin韋伯分佈之兩物件串聯系統的可靠度分析

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姓名

周仕鎧(Se-Kai Ju) 查詢紙本館藏

畢業系所

統計研究所

論文名稱

壽命具二元Marshall-Olkin韋伯分佈之兩物件串聯系統的可靠度分析
(Reliability Analysis of Two-Component Series System under Marshall-Olkin Bivariate Weibull Lifetime Distribution)

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摘要(中)

在串聯系統中，通常假設系統物件是彼此獨立的。本文考慮在型一設限下，以二元 Marshall-Olkin 韋伯分布去配適兩物件的壽命。而在串聯系統中，時常無法觀測到造成系統失效的物件，此時該系統為隱蔽資料。當系統壽命被設限或隱蔽時，資料中具遺失值，故此，我們使用期望值-最大化演算法去求得模型中未知參數的最大概似估計。同時，我們也利用遺失資訊法則去計算最大概似估計的費雪訊息矩陣，以近似最大概似估計的標準誤差。更進一步地我們將其分別推廣到定應力和階段應力加速壽命試驗，並比較各模型中所得之模型參數估計及系統與個別物件的平均壽命、分位數和可靠度函數之統計推論。由模擬結果顯示，我們所提出的方法頗為精確，另外我們也成功地分析了一組真實資料。

摘要(英)

Problems in series systems usually assume the lifetime distributions of components are independent. In this thesis, we consider to model the lifetime of a twocomponent series system under Type-1 censoring scheme by the Marshall-Olkin bivariate Weibull distribution. It is often to include masked data in which the component that causes the system to fail is not observed. When the data are masked or censored, there exist missing variables in the model. We apply the EM-algorithm to find the MLEs of the unknown parameters. In addition, we calculate the Fisher information via missing information principle to approximate the standard errors of the MLEs. Furthermore, we extend these results to the constant and step-stress accelerated life tests. Statistical inference on the lifetime distribution as the mean lifetimes, reliability functions and the quantiles of system and components are derived. Simulation study shows that the proposed methods perform accurately. A real data set is analyzed successfully.

關鍵字(中)

★ 串聯系統
★ 型一設限
★ 隱蔽資料
★ 期望值-最大化演算法
★ 遺失資訊法則
★ 加速壽命試驗

關鍵字(英)

★ Series System
★ Type-I censoring Scheme
★ Masked data
★ EM algorithm
★ Missing Information Principle
★ Accelerated Life Test

論文目次

Table of Contents
1 Introduction 6
1.1 Motivation and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Proposed Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Life Test of a Series System under Marshall-Olkin Bivariate Weibull Distribution
13
2.1 The Marshall-Olkin Bivariate Weibull Distribution . . . . . . . . . . . . . . . . 13
2.2 Series System under The Marshall-Olkin Bivariate Weibull Distribution . . . . . 18
2.3 Maximum Likelihood Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 Unequal Shape Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.2 Common Shape Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 The Asymptotic Variance-Covariance Matrix . . . . . . . . . . . . . . . . . . . . 31
2.5 Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 Accelerate Life Test of the MOBW with Common Shape Parameter 51
3.1 Constant-Stress Accelerate Life Test . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 Maximum Likelihood Estimate of CSALT . . . . . . . . . . . . . . . . . 51
3.1.2 The Asymptotic Variance-Covariance Matrix of CSALT . . . . . . . . . . 54
3.2 Step-Stress Accelerate Life Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.1 Maximum Likelihood Estimate of SSALT . . . . . . . . . . . . . . . . . . 57
3.2.2 The Asymptotic Variance-Covariance Matrix of SSALT . . . . . . . . . . 60
3.3 Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Simulation Study 68
4.1 Life Tests under MOBW Distribution . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Accelerated Life Test under MOBW Model . . . . . . . . . . . . . . . . . . . . . 78
4.3 Real Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 Conclusion 92
Reference 93
Appendix 97
3

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指導教授

樊采虹(Tasi-Hung Fan)

審核日期

2013-7-22

推文