Accelerated Life Testing of One-shot Devices Data Collection and Analysis

Provides authoritative guidance on statistical analysis techniques and inferential methods for one-shot device life-testing

Estimating the reliability of one-shot devices—electro-expolsive devices, fire extinguishers, automobile airbags, and other units that perform their function only once—poses unique analytical challenges to conventional approaches. Due to how one-shot devices are censored, their precise failure times cannot be obtained from testing. The condition of a one-shot device can only be recorded at a specific inspection time, resulting in a lack of lifetime data collected in life-tests.

Accelerated Life Testing of One-shot Devices: Data Collection and Analysis addresses the fundamental issues of statistical modeling based on data collected from accelerated life-tests of one-shot devices. The authors provide inferential methods and procedures for planning accelerated life-tests, and describe advanced statistical techniques to help reliability practitioners overcome estimation problems in the real world. Topics covered include likelihood inference, competing-risks models, one-shot devices with dependent components, model selection, and more. Enabling readers to apply the techniques to their own lifetime data and arrive at the most accurate inference possible, this practical resource:

• Provides expert guidance on comprehensive data analysis of one-shot devices under accelerated life-tests
• Discusses how to design experiments for data collection from efficient accelerated life-tests while conforming to budget constraints
• Helps readers develops optimal designs for constant-stress and step-stress accelerated life-tests, mainstream life-tests commonly used in reliability practice
• Includes R code in each chapter for readers to use in their own analyses of one-shot device testing data
• Features numerous case studies and practical examples throughout
• Highlights important issues, problems, and future research directions in reliability theory and practice

NARAYANASWAMY BALAKRISHNAN, PhD, is Distinguished University Professor, Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada.

MAN HO LING, PhD, is Associate Professor, Department of Mathematics and Information Technology, The Education University of Hong Kong, Hong Kong SAR, China.

HON YIU SO is Post-Doctoral Fellow, University of Waterloo, Waterloo, Ontario, Canada.

Preface xv

1 One-Shot Device Testing Data 1

1.1 Brief Overview 1

1.2 One-Shot Devices 1

1.3 Accelerated Life-Tests 4

1.4 Examples in Reliability and Survival Studies 7

1.4.1 Electro-explosive devices data 7

1.4.2 Glass capacitors data 7

1.4.3 Solder joints data 8

1.4.4 Grease-based magnetorheological fluids data 9

1.4.5 Mice tumor toxicological data 9

1.4.6 ED01 experiment data 10

1.4.7 Serial sacrifice data 11

1.5 Recent Developments in One-Shot Device Testing Analysis 11

2 Likelihood Inference 17

2.1 Brief Overview 17

2.2 Under CSALTs and Different Lifetime Distributions 18

2.3 EM-Algorithm 19

2.3.1 Exponential distribution 21

2.3.2 Gamma distribution 24

2.3.3 Weibull distribution 29

2.4 Interval Estimation 35

2.4.1 Asymptotic confidence intervals 35

2.4.2 Approximate confidence intervals 39

2.5 Simulation Studies 41

2.6 Case Studies with R Codes 52

3 Bayesian Inference 57

3.1 Brief Overview 57

3.2 Bayesian Framework 57

3.3 Choice of Priors 59

3.3.1 Laplace prior 60

3.3.2 Normal prior 60

3.3.3 Beta prior 62

3.4 Simulation Studies 63

3.5 Case Study with R Codes 72

4 Model Mis-Specification Analysis and Model Selection 77

4.1 Brief Overview 77

4.2 Model Mis-Specification Analysis 78

4.3 Model Selection 79

4.3.1 Akaike information criterion 79

4.3.2 Bayesian information criterion 81

4.3.3 Distance-Based Test Statistic 82

4.3.4 Parametric bootstrap procedure for testing goodness-of-fit 85

4.4 Simulation Studies 86

4.5 Case Study with R Codes 94

5 Robust Inference 97

5.1 Brief Overview 97

5.2 Weighted Minimum Density Power Divergence Estimators 98

5.3 Asymptotic Distributions 101

5.4 Robust Wald-type Tests 101

5.5 Inuence Function 103

5.6 Simulation Studies 106

5.7 Case Study with R Codes 110

6 Semi-Parametric Models and Inference 117

6.1 Brief Overview 117

6.2 Proportional Hazards Models 117

6.3 Likelihood Inference 121

6.4 Test of Proportional Hazard Rates 123

6.5 Simulation Studies 124

6.6 Case Studies with R Codes 128

7 Optimal Design of Tests 131

7.1 Brief Overview 131

7.2 Optimal Design of CSALTs 131

7.3 Optimal Design with Budget Constraints 133

7.3.1 Subject to specified budget and termination time 134

7.3.2 Subject to standard deviation and termination time 135

7.4 Case Studies with R Codes 136

7.5 Sensitivity of Optimal Designs 145

8 Design of Simple Step-Stress Accelerated Life-Tests 151

8.1 Brief Overview 151

8.2 One-Shot Device Testing Data under Simple SSALTs 151

8.3 Asymptotic Variance 154

8.3.1 Exponential distribution 154

8.3.2 Weibull distribution 156

8.3.3 With a known shape parameter w2 159

8.3.4 With a known parameter about stress level w1 160

8.4 Optimal Design of Simple SSALT 162

8.5 Case studies with R codes 165

8.5.1 SSALT for exponential distribution 165

8.5.2 SSALT for Weibull distribution 169

9 Competing-Risks Models 181

9.1 Brief Overview 181

9.2 One-Shot Device Testing Data with Competing Risks 181

9.3 Likelihood Estimation for Exponential Distribution 184

9.3.1 Without masked failure modes 185

9.3.2 With masked failure modes 190

9.4 Likelihood Estimation for Weibull Distribution 193

9.5 Bayesian Estimation 201

9.5.1 Without masked failure modes 201

9.5.2 Laplace prior 203

9.5.3 Normal prior 204

9.5.4 Dirichlet prior 205

9.5.5 With masked failure modes 207

9.6 Simulation Studies 207

9.7 Case Study with R Codes 215

10 One-Shot Devices with Dependent Components 223

10.1 Brief Overview 223

10.2 Test Data with Dependent Components 223

10.3 Copula Models 224

10.3.1 Family of Archimedean copulas 226

10.3.2 Gumbel-Hougaard copula 227

10.3.3 Frank copula 231

10.4 Estimation of Dependence 234

10.5 Simulation Studies 236

10.6 Case Study with R Codes 238

11 Conclusions and Future Directions 245

11.1 Brief Overview 245

11.2 Concluding Remarks 245

11.2.1 Large sample sizes for flexible models 245

11.2.2 Accurate estimation 246

11.2.3 Good designs before data analysis 247

11.3 Future Directions 248

11.3.1 Weibull lifetime distribution with threshold parameter 248

11.3.2 Frailty models 248

11.3.3 Optimal design of SSALTs with multiple stress levels 249

11.3.4 Comparison of CSALTs and SSALTs 249

A Derivation of Hi(a; b) 251

B Observed Information Matrix 253

C Non-Identifiable Parameters for SSALTs under Weibull Distribution 257

D Optimal Design under Weibull Distributions with Fixed w1 259

E Conditional Expectations for Competing Risks Model under Exponential Distribution 261

F Kendall's Tau for Frank Copula 267

Index 287

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