Reliability, Life Testing and the Prediction of Service Lives: For Engineers and Scientists (Springer Series in Statistics)

0387325220......Page 1
Reliability, Life Testing\rand the Prediction of\rService Lives......Page 4
Copyright Page......Page 5
Preface......Page 6
Vörtrekkers......Page 8
Glossary......Page 9
Admonitions......Page 10
Table of Contents......Page 12
1.1. Why Reliability Is Important......Page 16
1.2.1. Concepts from Probability......Page 18
1.2.2. Concepts from Statistics......Page 21
2.1. Properties of Life Distributions......Page 25
2.2.1. The Epstein (Exponential) Distribution......Page 29
2.2.3. The Pareto Distribution......Page 30
2.2.4. The Gaussian or Normal Distribution......Page 31
The Truncated Normal Distribution......Page 32
The Log-Normal Distribution......Page 33
The Xi-Normal Family......Page 34
2.2.7. The Inverse-Gaussian Distribution......Page 35
2.2.8. The Extreme-Value Distribution of Minima......Page 36
2.2.9. Some Other Distributions......Page 37
3.1. Binomial Coefficients and Sterling Numbers......Page 41
3.1.1. Lagrange Coefficients......Page 43
3.2.1. Lotteries......Page 45
3.2.2. Coupon Collecting......Page 46
3.3.1. Occupancy......Page 49
Multiple Occupancy......Page 51
3.3.2. Allocations......Page 53
3.4.1. The Sum of Epstein Waiting Times......Page 54
3.4.2. Interpolation and Numerical Integration......Page 55
4.1. Functional Representation......Page 59
4.2. Event-Tree Depiction......Page 65
4.2.1. Associated Random Variables......Page 66
4.3. Evaluation of Reliability......Page 68
4.3.1. System Life......Page 70
4.4. Use of Association to Bound Reliability......Page 75
4.5. Shape of the Reliability Function......Page 78
4.6.2. Diagnostics Using Reliability......Page 81
4.7. Hazard Rates and Pόlya Frequency Functions......Page 83
4.8. Closure Properties......Page 84
4.8.1. Further Closure Properties......Page 86
5.1. The Gaussian or Normal Distribution......Page 90
5.2. Epstein\'s Distribution......Page 92
5.3.1. The Log-Normal Distribution......Page 93
5.3.2. The Fatigue-Life Distribution......Page 94
5.4. Discovery and Rediscovery......Page 95
5.5.1. Gumbel\'s Theory......Page 97
5.5.2. Maximum Loads and Association......Page 99
6.1. Likelihood without Priors......Page 104
6.2. Likelihood for Complete Samples......Page 107
6.3.1. The Likelihood Depends upon the Model......Page 109
6.3.2. Relative Likelihoods Are Not Probabilities on Θ......Page 110
6.3.4. Likelihoods on Simple Parameter Spaces......Page 111
6.3.5. Bayes\' Theorem and Its Application......Page 113
6.4. Types of Censoring of Data......Page 116
6.4.2. Estimation for Type II Censoring......Page 117
6.4.3. Estimation for Type III Random Censoring......Page 118
6.4.4. Transformation to the Standard Weibull......Page 119
6.5. Generation of Ordered Observations......Page 120
6.6. A Parametric Model of Censoring......Page 123
6.7. The Empirical Cumulative Distribution......Page 126
7.1.1. Life-Table Methods......Page 129
7.1.2. The Kaplan-Meier Estimator......Page 130
7.2. Expectation and Bias of the K-M Estimator......Page 132
Proportional Hazards......Page 135
7.3. The Variance and Mean-Square Error......Page 137
7.4. The Nelson-Aalen Estimator......Page 139
7.4.1. Extensions and Generalizations......Page 140
8.2. Estimation for the Weibull Distribution......Page 143
8.2.2. Left-Censored Estimation......Page 144
8.3.1. The Bathtub-Shaped Hazard......Page 145
8.4.1. Estimation under Independent Competing Risks......Page 146
8.4.2. Observing Both Time and Cause of Failure......Page 147
8.4.3. Estimation with Dependent Failure Modes......Page 149
8.4.4. Estimation under Random Censoring on Both Sides......Page 150
8.4.5. Censoring for the Reciprocal Weibull......Page 152
8.5. Change Points and Multiple Failure Mechanisms......Page 154
8.5.1. A Known Change Point......Page 155
8.5.2. A Change Point at an Unknown Location......Page 159
8.5.3. Conclusions......Page 161
9.1. Scientific Idealism......Page 163
9.2. Consultation and Diagnosis......Page 164
9.3. Datasets in Service-Life Prediction......Page 166
9.4. Data, Consulting, and Modeling......Page 172
10.1. The Past as Prologue......Page 175
10.2. The Fatigue-Life Distribution......Page 177
10.3. The Mixed Class of Cumulative Damage Distributions......Page 179
10.4. Elementary Derivation of Means and Variances......Page 181
10.5. Behavior of the Hazard Rate......Page 183
10.6. Mixed Variate Relationships......Page 187
10.7.1. Estimation for Complete Samples......Page 191
Estimation of β When α Is Known......Page 192
Unbised Estimation......Page 193
10.7.2. Estimation for Incomplete Wald Samples......Page 195
10.8.1. Complete Samples......Page 197
10.8.2. Incomplete Samples of Fatigue-life Distribution......Page 199
10.9. Estimation for Tweedie\'s Distribution......Page 202
10.10.2. When the CD-Distributions Are Unknown......Page 204
10.10.3. Weibull Distribution Contrasted with the FL-Distribution......Page 205
10.10.4. Galton Distribution Mistaken for FL-Distribution......Page 206
11.1. Applicability......Page 209
11.3. Sample Distributions under Consonance......Page 210
11.3.1. And Student\'s Distribution?......Page 219
11.4. Classifications for Dispersion Analysis......Page 221
11.4.1. A Single Classification......Page 222
No Row or Column Effects......Page 223
No Column Effects......Page 224
When Does Consonance Occur?......Page 226
12.1. The Poisson Process......Page 229
12.2. Damage Due to Intermittant Shocks......Page 231
12.3. Renewal Processes......Page 234
12.3.1. Renewal Function for the Wald Distribution......Page 236
12.3.3. Tauberian Theorems for the Laplace Transform......Page 238
12.4. Shock Models with Varying Intensity......Page 239
12.4.1. The Marshall-Olkin Distribution......Page 240
12.5. Stationary Renewal Processes......Page 242
12.6. The Miner-Palmgren Rule and Additive Damage......Page 244
12.6.1. Miner\'s Rule as an Expectation......Page 245
12.6.2. How Applicable Is This Theory?......Page 246
12.7.1. Deterioration of Polymer Coatings......Page 247
12.7.2. Varying Duty Cycles......Page 248
12.8. When Linear Cumulative Damage Fails......Page 250
12.8.1. Load-Order Effects in Crack Propagation......Page 251
13.1. Wear under Spectral Loading......Page 255
13.2. Multivariate Fatigue Life......Page 256
13.2.1. Two-Component Load Sharing......Page 257
13.2.2. The Multivariate Fatigue-Life Distribution......Page 258
13.3. Correlations between Component Damage......Page 263
13.3.1. Covariance and Association......Page 264
13.4. Implementation......Page 268
13.4.1. Estimation for Small Censored Samples......Page 269
13.4.2. Relating Cumulative-Damage Parameters to the Exposure......Page 270
14.1.1. Introduction......Page 273
Static Strength for a Column......Page 274
14.1.3. Strength of an Airframe Subsystem......Page 276
14.2. Accelerated Tests for Strength......Page 277
14.2.1. Determination of the Part of Least Accord......Page 279
14.3. Danger of Extrapolation from Tests......Page 281
The Pagett Models Using the Wald Distribution......Page 283
14.4. Fracture Mechanics and Stochastic Damage......Page 285
15.2. Availability......Page 288
15.2.1. Application of Tauberian Properties......Page 290
15.2.2. System Availability......Page 291
15.3. Age Replacement with Renewal......Page 292
15.3.1. A Single Machine with Repair......Page 294
15.4. The Inversion of Transforms......Page 296
15.5. Problems in Scheduled Maintenance......Page 299
15.5.1. A Problem with Unscheduled Fleet Maintenance......Page 300
15.5.2. A Problem with Scheduled Fleet Maintenance......Page 301
16.1.1. Stieltjes Integrals......Page 304
16.2. Probability and Measure......Page 306
16.3. Distribution Transforms......Page 308
16.4. A Compendium of Discrete Distributions......Page 312
16.5. A Compendium of Continuous Distributions......Page 313
Bibliography......Page 314
Index......Page 320