Mathematical and Statistical Models and Methods in Reliability (eBook)

Contents 6
Preface 16
List of Contributors 18
List of Tables 22
List of Figures 24
Reliability Models, Methods, and Optimization 28
1 Reliability of Semi-Markov Systems with Asymptotic Merging Phase Space 29
1.1 Introduction 29
1.2 Reliability of the Renewal System 30
1.3 Absorbing Time of Semi-Markov Process 33
1.4 Reward Functional in the Semi-Markov Environment 37
1.5 Dynamic Reward Functional 38
1.6 Fluctuations of the Reward Functional 39
1.7 Heuristic Phase Merging 39
1.8 Stationary Phase Merging Scheme 40
References 43
2 Nonlinearly Perturbed Stochastic Processes and Systems 45
2.1 Introduction 45
2.2 Nonlinearly Perturbed Renewal Equation 47
2.3 Nonlinearly Perturbed Regenerative Processes 53
2.4 Nonlinearly Perturbed Semi-Markov Processes 56
2.5 Nonlinearly Perturbed Stochastic Systems 57
2.6 Nonlinearly Perturbed Risk Processes 58
2.7 Conclusion 59
References 60
3 On a Copula for Failure Times of System Elements 64
3.1 Introduction 64
3.2 Multidimensional Distribution 66
3.3 Different Properties 68
3.4 Copula 71
3.5 Example 72
3.6 Conclusion 73
3.7 Appendix 74
References 75
4 On One Method of Reliability Coefficients Calculation for Objects in Non- Homogeneous Event Flows 76
4.1 Introduction 76
4.2 General Concepts 77
4.3 Review of Models Taking into Account Non-Homogeneity 77
4.4 Normalizing Flow Function Model 78
4.5 Leading Flow Function and Failure Flow Parameter 80
4.6 Getting of Distribution Law for the Operability Cycle 81
4.7 Getting of Resource Reliability Characteristics 83
4.8 Availability Function Calculation 85
4.9 Numerical Computation of Availability Function 89
References 90
Appendix 91
5 A New Approach to Maintenance Optimization by Modeling Intensity Control 93
5.1 Introduction 94
5.2 Model 95
5.3 Modelling Intensity Control Given Partial Information 100
5.4 Numerical Example 104
References 108
6 Longitudinal Latent Markov Processes Observable Through an Invariant Rasch Model 110
6.1 Introduction 110
6.2 Description of the Latent Process 111
6.3 Application to Markov Latent Processes 114
6.4 Estimation 118
6.5 Estimation of Rasch Latent Markov Processes 120
References 122
7 Dynamics of Dependence Properties for Lifetimes Influenced by Unobservable Environmental Factors 124
7.1 Introduction 124
7.2 Some Basic Facts 127
7.3 Monotonicity Properties of Survival Copulas 129
7.4 Concluding Remarks 133
References 134
Appendix 134
8 On Alternative of Choice for a Prophylaxis Problem 136
8.1 Renewal System 136
8.2 Optimization Problem 138
8.3 Alternative of Choice for Twice Regenerative System 139
References 145
9 Optimal Incomplete Maintenance for Systems with Discrete Time- to- Failure Distribution 146
9.1 Introduction 146
9.2 Discrete Lifetime Distributions and Their Failure Rate 146
9.3 Kijima Type Repairs 148
9.4 Optimal Maintenance as Time Scale Transformation 149
9.5 Simulation of Failure–Repair Processes 151
9.6 Cost Optimal Maintenance 153
9.7 Conclusion 155
References 155
10 A Gini-Type Index for Aging/Rejuvenating Objects 156
Acronyms 156
10.1 Introduction 156
10.2 GT Coefficient for Repairable Systems 158
10.3 GT Coefficient for Non-repairable Systems (Components) 160
10.4 Conclusions 162
References 163
11 Redundancy Analysis for Multi-state System: Reliability and Financial Assessment 164
11.1 Introduction 164
11.2 Problem Formulation 166
11.3 Model Descriptions 168
11.4 Algorithm of UGF Computation for Entire MSS 176
11.5 Reliability Measures Computation for Entire MSS 177
11.6 Illustrative Example 177
11.7 Conclusions 179
References 179
12 On the Reliability Modeling of Hierarchical Systems 181
12.1 Introduction and Motivation 181
12.2 General Model 182
12.3 Investigation of the Reliability of a General Model 183
12.4 The Subsystems Behavior Investigation 184
12.5 An Example 193
12.6 Further Problems 194
References 195
Statistical Methods in Reliability 197
13 Parametric Estimation of Redundant System Reliability From Censored Data 198
13.1 Introduction 198
13.2 Parametric Point Estimators of the c.d.f. Kj( t) 199
13.3 Asymptotic Confidence Intervals for Kj( t) 200
References 212
14 Assessing Accuracy of Statistical Inferences by Resamplings 213
14.1 Introduction 213
14.2 Linear Heteroscedastic Regression Model 215
14.3 Resampling from Estimates of Residuals 217
14.4 OLS-Estimators of Regression Coefficients and Their Accuracies 219
14.5 Conclusion 221
References 223
Appendix 224
15 Change Point Estimation in Regression Models with Fixed Design 227
15.1 Introduction 227
15.2 The Jump Case 230
15.3 The Continuous Case 236
References 239
Appendix 239
16 A Model for Field Failure Prediction Using Dynamic Environmental Data 242
16.1 Introduction 242
16.2 Data and Model 244
16.3 Parameter Estimation 247
16.4 Predictions 248
16.5 Calibration of Prediction Intervals 249
16.6 Conclusions and Areas for Future Research 250
References 251
17 Efficient Regression Estimation Under General Censoring and Truncation 253
17.1 Framework 253
17.2 Data Description 254
17.3 The Semi-Parametric Model 255
17.4 Inference About ß 258
17.5 Perpectives and References 259
References 259
18 On Generalized Tests of Fit for Multinomial Populations 260
18.1 Introduction 260
18.2 The F-Family of Test Statistics 263
18.3 Empirical Distribution and Power Simulations 264
18.4 Conclusions 269
References 270
19 Modeling and Scaling of Categorical Data 272
19.1 Introduction 272
19.2 Basic Model 273
19.3 Criteria for Scaling 277
19.4 Calculation of Most Separating Scales 279
References 281
20 Nonparametric Estimation and Testing the Effect of Covariates in Accelerated Life Time Models Under Censoring 282
20.1 Introduction 282
20.2 Nonparametric Estimation of the Regression Function Under Censoring 284
20.3 Properties of the Nonparametric Regression Estimator 286
20.4 Formulation of the Test Procedure 291
References 293
21 Nonparametric Estimation of Time Trend for Repairable Systems Data 294
21.1 Introduction 294
21.2 Definitions and Preliminaries 295
21.3 A Motivating Example 297
21.4 Nonparametric Estimation of a Monotone .(·) 298
21.5 Kernel-Estimators for .(t) in the General Case 299
21.6 Concluding Remarks 303
References 305
22 Confidence Region for Distribution Function from Censored Data 306
22.1 Introduction 306
22.2 Testing of the Homogeneity Hypothesis 311
References 312
23 Empirical Estimate with Uniformly Minimal d-Risk for Bernoulli Trials Success Probability 313
23.1 Introduction 313
23.2 An Estimate with Uniformly Minimal d-Risk 314
23.3 Empirical Estimate Based on the Modification of the Historical Data 316
23.4 Empirical Estimate Based on the Invariance of EUMD for the Scalar Parameter of a Discrete Exponential Family 319
23.5 Accuracy Properties Investigation for the Estimates Based on the Statistical Modeling 321
References 322
24 Estimation of Archival Lifetime Distribution for Writable Optical Disks from Accelerated Testings 323
24.1 Introduction 323
24.2 Models 323
24.3 Estimation of Parameters and Percentiles 325
24.4 An Example 327
24.5 Concluding Remarks 328
References 328
Applications 330
25 Ages in Reliability and Bio Systems, Interpretations, Control, and Applications 331
25.1 Introduction 331
25.2 Main Definitions and Equivalent Representations 332
25.3 Empirical Distribution Functions and Empirically Equivalent Ages 335
25.4 Equivalent Ages Have Equivalent/Equal Accumulated Stress: Comparing Ages of Processes 336
25.5 Multidimensional Life Time Equivalence 343
25.6 Conclusions 347
References 347
26 Shocks in Mixed Populations 349
26.1 Introduction 349
26.2 Two Ordered Subpopulations 350
26.3 Continuous Mixtures 353
26.4 Concluding Remarks 357
References 357
27 Bayesian Estimation of Degradation Model Defined by a Wiener Process 359
27.1 Introduction 359
27.2 Reliability Testing 360
27.3 Bayesian Estimation of Wiener Process Parameters 362
27.4 Estimation of Prior Mean of Wiener Process by FEM Method: Application to Brake Disc Wear 366
27.5 Results 368
27.6 Conclusion 370
References 371
28 Benefits of Threshold Regression: A Case-Study Comparison with Cox Proportional Hazards Regression 372
28.1 Introduction 372
28.2 First-Hitting Time (FHT) and Threshold Regression (TR) Model 373
28.3 Threshold Regression (TR) Investigations of Lung Cancer 375
28.4 Comparisons of Results Obtained from the Cox PH Model 378
References 382
29 Optimal Stopping and Reselling of European Options 384
29.1 Introduction 384
29.2 Formulation of the Reselling Problem 385
29.3 Convergence of Rewards for Multivariate Markov Price Processes 387
29.4 Binomial–Trinomial Approximations for Reselling Model 391
29.5 Numerical Examples 398
References 400
30 Bayesian Modeling of Health State Preferences 403
30.1 Introduction 403
30.2 Bayesian Modeling of Attribute Utilities 405
30.3 Modeling Attribute Weights 407
30.4 Bayesian Evaluation of Health States 409
30.5 Concluding Remarks 410
References 410
31 Information Measures in Biostatistics and Reliability Engineering 412
31.1 Introduction 412
31.2 Modelling in Biomedicine and Reliability 413
31.3 Applications 420
References 422
Reliability Computer Tools 425
32 Software System for Simulation and Research of Probabilistic Regularities and Statistical Data Analysis in Reliability and Quality Control 426
32.1 Introduction 426
32.2 The Investigation of Parameter Estimates Properties 427
32.3 The Investigation of Nonparametric Goodness-of-Fit Test Statistic Distributions 427
32.4 The Investigation of Statistic Distributions and the Power of .2 Tests 429
32.5 The Comparative Analysis of the Power of Goodness- of- Fit Tests 432
32.6 The Investigation of Statistic Distributions and the Power of Normality Tests 433
32.7 The Investigation of Homogeneity Test Statistic Distributions 433
32.8 The Investigation of Statistic Distributions and the Power of Tests for Homogeneity of Means 434
32.9 The Investigation of Statistic Distributions and the Power of Tests for Homogeneity of Variances 435
32.10 Conclusion 437
References 438
33 Inverse Gaussian Model and Its Applications in Reliability and Survival Analysis 442
33.1 Introduction 442
33.2 The Family of the Inverse Gaussian Distributions 443
33.3 Goodness-of-Fit Tests for the Family of IGD 446
33.4 About Testing Hypotheses for the Inverse Gaussian Distribution 448
33.5 Chi-Squared Goodness-of-fit Test for the Family of IGD in Case of Censored Data 454
33.6 Models with Covariates Based on the Family of IGD 456
References 461
Index 463