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Bayesian Designing and Analysis of Simple Step-Stress Accelerated Life Test With Bayesian Designing and Analysis of Simple Step-Stress Accelerated Life Test with Weibull Lifetime Distribution A thesis presented to the faculty of the Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirements for the degree Master of Science Xi Liu November 2010 © 2010 Xi Liu. All Rights Reserved. 2 This thesis titled Bayesian Designing and Analysis of Simple Step-Stress Accelerated Life Test with Weibull Lifetime Distribution by XI LIU has been approved for the Department of Industrial and Systems Engineering and the Russ College of Engineering and Technology by Tao Yuan Assistant Professor of Industrial and Systems Engineering Dennis Irwin Dean, Russ College of Engineering and Technology 3 ABSTRACT LIU, XI, M.S., November 2010, Industrial and Systems Engineering Bayesian Designing and Analysis of Simple Step-Stress Accelerated Life Test with Weibull Lifetime Distribution (82 pp.) Director of Thesis: Tao Yuan This thesis develops a Bayesian method to analyze and plan a simple step-stress accelerated life test with Type-I censoring. It is assumed that the failure times at each stress level follow the Weibull distribution and the scale parameter is a log-linear function of the stress level. The point estimates of model parameters both by ML analysis and Bayesian analysis are organized in the thesis. The asymptotic confidence intervals and Bayesian confidence intervals are deduced. The influence of prior distribution on the estimation is studied. The Bayesian optimal plan is obtained by minimizing the pre- posterior variance of a specified low percentile of the life time distribution at the normal stress condition. Monte Carlo simulation algorithm based on Gibbs sampling is developed to find the optimal stress changing time. The maximum likelihood method is also developed for the use of comparison. Influence of prior distribution and sample size on the optimal plan is also discussed. The results show that Bayesian approach has hopeful potential in analyzing and designing reliability life test when there is prior knowledge and uncertainty for the model parameters. Approved: _____________________________________________________________ Tao Yuan Assistant Professor of Industrial and Systems Engineering 4 ACKNOWLEDGMENTS During the development of my graduate studies in Ohio University, several persons and institutions collaborated directly and indirectly with my research. Without their support, it would be impossible for me to finish my work. That is why I wish to dedicate this section to recognize their support. First and foremost I offer my sincerest gratitude to my advisor, Dr. Tao Yuan, who has supported and guided me throughout my thesis with his patience and knowledge. Without his encouragement and effort this thesis would not have been completed. Besides my advisor, I would like to thank the rest of my thesis committee members: Dr. Andrew Snow, Dr. David Koonce and Dr. Diana Schwercha for their guidance, insightful comments, and suggestions. I would like to thank Saleem Z. Ramadan for supplying me data set as my case study. The experimental data make the thesis more practical. This work was partially supported by National Science Foundation project CMMI-092642. Finally, I would like to thank my family, for their unconditional support, inspiration, and love. 5 TABLE OF CONTENTS ABSTRACT ........................................................................................................................ 3 ACKNOWLEDGMENTS .................................................................................................. 4 TABLE OF CONTENTS .................................................................................................... 5 LIST OF TABLES .............................................................................................................. 7 LIST OF FIGURES ............................................................................................................ 8 CHAPTER 1: INTRODUCTION ....................................................................................... 9 1.1 Motivation ................................................................................................................. 9 1.2 Background ............................................................................................................... 9 1.2.1 Reliability ........................................................................................................... 9 1.2.2 Accelerated life test .......................................................................................... 12 1.2.3 Step-stress accelerated life test ........................................................................ 12 1.2.4 The cumulative exposure model ...................................................................... 14 1.2.5 Acceleration models ......................................................................................... 16 1.2.6 Maximum likelihood estimation (MLE) .......................................................... 17 1.2.7 Bayesian inference ........................................................................................... 18 1.3 Objectives ............................................................................................................... 20 CHAPTER 2: LITERATURE REVIEW ......................................................................... 22 2.1 Analysis of Simple SSALT ..................................................................................... 22 2.1.1 Analysis of simple SSALT using ML method ................................................. 22 2.1.2 Bayesian analysis of simple SSALT ................................................................ 23 2.2 Design of Simple SSALT ....................................................................................... 24 2.2.1 Determination of optimal stress changing time ............................................... 24 2.2.2 Determination of optimal sample size ............................................................. 26 CHAPTER 3: PROBLEM STATEMENT........................................................................ 27 3.1 Notations ................................................................................................................. 27 3.2 Basic Assumptions .................................................................................................. 27 3.3 Problem Description ............................................................................................... 28 3.3.1 Analysis of simple SSALT .............................................................................. 28 3.3.2 Design of simple SSALT ................................................................................. 29 3.4 Development of ML Methods ................................................................................. 29 6 3.5 Development of Simulation Algorithm .................................................................. 29 3.6 Investigation of Practical Approach to Specifying the Prior Distribution .............. 30 3.7 Comparison between ML method and Bayesian method ....................................... 30 3.8 Sensitivity Analysis ................................................................................................ 30 CHAPTER 4: METHODOLIGIES................................................................................... 32 4.1 Analysis of SSALT ................................................................................................. 32 4.1.1 ML analysis ...................................................................................................... 32 4.1.2 Bayesian analysis ............................................................................................. 35 4.2 Optimal Design of Simple SSALT ......................................................................... 37 4.2.1 Maximum likelihood plan ................................................................................ 37 4.2.2 Bayesian plan ................................................................................................... 39 4.2.3 Simulation algorithm ....................................................................................... 41 4.2.4 Prior specification ............................................................................................ 42 CHAPTER 5: CASE STUDY ........................................................................................... 44 5.1 Analysis of Simple SSALT ..................................................................................... 44 5.2 Design of Simple SSALT ....................................................................................... 50 CHAPTER 6: CONCLUSIONS AND FUTURE RESEARCH ....................................... 57 6.1 Conclusions ............................................................................................................. 57 6.1.1 Analysis of SSALT .......................................................................................... 58 6.1.2 Design of SSALT ............................................................................................. 58 6.2 Future Research ...................................................................................................... 59 REFERENCES ................................................................................................................. 61 APPENDIX A: OBSERVED FISHER INFORMATION MATRIX ............................... 65 APPENDIX B: INFERENCE OF MAXIMUM LIKELIHOOD
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