Bayesian design for life tests and accelerated life tests

114 Pages·2016·2.1 MB·English

by  Yao Zhang

Iowa State University Capstones, Theses and
Retrospective Theses and Dissertations
Dissertations
2002
Bayesian design for life tests and accelerated life
tests
Yao Zhang
Iowa State University
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Part of theStatistics and Probability Commons
Recommended Citation
Zhang, Yao, "Bayesian design for life tests and accelerated life tests " (2002).Retrospective Theses and Dissertations. 493.
https://lib.dr.iastate.edu/rtd/493
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Bayesian design for life tests and accelerated life tests
by
Yao Zhang
A dissertation submitted to the graduate faculty
in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY
Major: Statistics
Program of Study Committee:
William Q. Meeker, Major Professor
David A. Carter-Lewis
Max Morris
Hal S. Stern
Huaiqing Wu
Iowa State University
Ames, Iowa
2002
Copyright © Yao Zhang, 2002. All rights reserved.
UMI Number: 3061878
UMT
UMI Microform 3061878
Copyright 2002 by ProQuest Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, Uniteid States Code.
ProQuest Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, Ml 48106-1346
ii
Graduate College
Iowa State University
This is to certify that the doctoral dissertation of
Yao Zhang
has met the dissertation requirements of Iowa State University
Signature was redacted for privacy.
Major Professor
Signature was redacted for privacy.
For the Major Program
iii
TABLE OF CONTENTS
1 GENERAL INTRODUCTION 1
1.1 Introduction 1
1.2 Dissertation Organization 3
2 BAYESIAN LIFE TEST PLANNING FOR THE WEIBULL DIS­
TRIBUTION WITH GIVEN SHAPE PARAMETER 5
Abstract 5
2.1 Introduction 6
2.1.1 Problem 6
2.1.2 Overview 7
2.2 Related Work 7
2.3 Planning Problem 9
2.3.1 Model and Bayes Estimation 9
2.3.2 Planning Based on Precision of a Quantile 10
2.4 Numerical Examples 13
2.4.1 Setup 13
2.4.2 Planning with the LSAPPF Criterion 14
2.4.3 Planning with the ERPCIL Criterion 17
2.4.4 Discussion 19
2.4.5 Comparison with Non-Bayesian Test Plans 20
2.5 Concluding Remarks and Areas for Future Research 22
iv
References 22
3 BAYESIAN DESIGN FOR LIFE TESTING 25
Abstract 25
3.1 Introduction 26
3.1.1 Background and Motivation 26
3.1.2 Related Work 27
3.1.3 Overview 28
3.2 The Bayesian Life Test Design Problem and The Criterion 29
3.3 General Large Sample Approximation for the Bayes Life Test Design Cri­
terion 31
3.3.1 The Posterior Distribution of Lifetime Model Parameters and Their
Variances 32
3.3.2 The Large Sample Approximate Bayes Criterion 33
3.4 Weibull Life Test Design with Prior Information on the Shape Parameter 34
3.5 Effects of Prior Information on the Posterior Variance of the Quantile of
Interest 36
3.6 Test Plan Solutions 37
3.7 Example 41
3.8 Discussion 44
3.8.1 Solution for Type I Censoring 44
3.8.2 Shape of Prior Distribution 44
3.8.3 Joint Informative Prior Information 45
3.9 Weibull Life Test with Given Shape Parameter 45
3.10 Simulation Approach 48
3.11 Comparison of the Simulation Approach and the Large Sample Approxi­
mation Approach 51
3.12 Concluding Remarks and Areas for Future Research 56
References 57
BAYESIAN OPTIMUM DESIGN FOR ACCELERATED LIFE TESTS 60
Abstract 60
4.1 Introduction 61
4.1.1 Background and Motivation 61
4.1.2 Related Work 62
4.1.3 Overview 64
4.2 The Bayesian ALT Design Problem 65
4.2.1 The Model 65
4.2.2 The Prior Distribution 66
4.2.3 The Bayes Design Criterion 67
4.3 Large-Sample Approximation for the Bayesian ALT Design Problem ... 68
4.3.1 Posterior Variance 68
4.3.2 The Large-Sample Approximate Bayes Criterion 70
4.4 General Equivalence Theorem for the Bayesian ALT Design Problem . . 71
4.5 Numerical Example 73
4.5.1 Description 73
4.5.2 Numerical Search for Optimum Design 75
4.5.3 Derivative Function at the Optimum Design 79
4.6 Further Investigation 79
4.6.1 Effect of Prior Information on the Slope Parameter 79
4.6.2 Effect of Censoring 83
4.6.3 Effect of Sample Size 84
4.6.4 Type II Censoring 85
4.7 Comparison with Non-Bayesian Optimum Design 86
vi
4.8 Simulation Evaluation for Bayesian ALT Design 87
4.8.1 Simulations Based on Fixed Model Parameters 88
4.8.2 Simulations Based on Simulated Model Parameters 88
4.9 Concluding Remarks and Areas for Future Research 91
References 93
5 CONCLUSION 96
APPENDIX A TECHNICAL DETAILS FOR RESULT (2.5) 98
APPENDIX B CONCAVITY OF THE BAYESIAN CRITERION (4.9)
FOR ALT DESIGN 99