The Art of Progressive Censoring (eBook)
XXI, 645 Seiten
Springer New York (Verlag)
978-0-8176-4807-7 (ISBN)
This book offers a thorough and updated guide to the theory and methods of progressive censoring, an area that has experienced tremendous growth over the last decade. The theory has developed quite nicely in some special cases having practical applications to reliability and quality.
The Art of Progressive Censoring is a valuable reference for graduate students, researchers, and practitioners in applied statistics, quality control, life testing, and reliability. With its accessible style and concrete examples, the work may also be used as a textbook in an advanced undergraduate or a beginning graduate course on censoring or progressive censoring, as well as a supplementary textbook for a course on ordered data.
This monograph offers a thorough and updated guide to the theory and methods of progressive censoring, an area that has experienced tremendous growth in recent years. Progressive censoring, originally proposed in the 1950s, is an efficient method of handling samples from industrial experiments involving lifetimes of units that have either failed or censored in a progressive fashion during the life test, with many practical applications to reliability and quality.Key topics and features:Data sets from the literature as well as newly simulated data sets are used to illustrate concepts throughout the textEmphasis on real-life applications to life testing, reliability, and quality controlDiscussion of parametric and nonparametric inferenceCoverage of experimental design with optimal progressive censoringThe Art of Progressive Censoring is a valuable reference for graduate students, researchers, and practitioners inapplied statistics, quality control, life testing, and reliability. With its accessible style and concrete examples, the work may also be used as a textbook in advanced undergraduate or beginning graduate courses on censoring or progressive censoring, as well as a supplementary textbook for a course on ordered data.
Preface 8
Section and Equation Numbering and Referencing 13
Contents 14
Part I Distribution Theory and Models 23
Chapter1 Progressive Censoring: Data and Models 24
1.1 Progressively Censored Data 25
1.1.1 Progressive Type-II Censoring 26
General Progressive Type-II Censoring 31
1.1.2 Progressive Type-I Censoring 31
Progressive Type-I Interval Censoring 35
1.1.3 Progressive Hybrid Censoring 35
Type-I Progressive Hybrid Censoring 36
Type-II Progressive Hybrid Censoring 38
1.2 Probabilistic Models in Progressive Censoring 40
Chapter2 Progressive Type-II Censoring: Distribution Theory 42
2.1 Joint Distribution 42
2.2 Connection to Generalized Order Statistics and Sequential Order Statistics 45
2.3 Results for Particular Population Distributions 47
2.3.1 Exponential Distributions 47
2.3.2 Reflected Power Distribution and UniformDistribution 52
2.3.3 Pareto Distributions 54
2.3.4 Progressive Withdrawal and Dual Generalized Order Statistics 54
2.4 Marginal Distributions 56
2.4.1 Exponential Distribution 57
2.4.2 Uniform Distribution 59
2.4.3 General Distributions 59
Multiply Censored Progressively Type-II Censored Order Statistics 61
An Important Recurrence Relation 62
An Alternative Approach to Derive the Marginals 63
Connection of Marginals to Interpolation Polynomials 64
2.5 Conditional Distributions 65
2.5.1 Markov Property 65
2.5.2 Distributions of Generalized Spacings 67
2.5.3 Block Independence of Progressively Type-II Censored Order Statistics 68
2.5.4 Dependence Structure of Progressively Type-II Censored Order Statistics 72
2.6 Basic Recurrence Relations 74
2.7 Shape of Density Functions 76
2.7.1 Log-Concavity of Uniform Progressively Type-II Censored Order Statistics 77
2.7.2 The Shape of Densities of Uniform Progressively Type-II Censored Order Statistics 78
2.7.3 Unimodality and Log-Concavity of Progressively Type-II Censored Order Statistics Based on F 80
2.8 Discrete Progressively Type-II Censored Order Statistics 82
2.9 Exceedances 86
Chapter3 Further Distributional Results on Progressive Type-II Censoring 88
3.1 Characterizations by Progressively Type-II Censored Order Statistics 88
3.1.1 Characterizations by Independence Properties 88
3.1.2 Characterizations by Distributional Properties 90
3.1.3 Characterizations via Regression 91
Conditional Expectations and Characterization Problems 92
Adjacent Progressively Type-II Censored Order Statistics 93
Progressively Type-II Censored Order Statistics Based on Higher-Order Gap 97
Reversed Regression of Adjacent Progressively Type-II Censored Order Statistics 98
3.1.4 Characterizations for Discrete Parents 99
Characterization by Regression 100
Characterization by Distribution Properties 101
Characterization by (Conditional) Independence 102
3.2 Stochastic Ordering of Progressively Type-II Censored Order Statistics 103
3.2.1 Univariate Stochastic Orders and Its Applications to Progressively Type-II Censored Order Statistics 103
Stochastic Order 103
Hazard Rate Order 105
Likelihood Ratio Order 106
Dispersive Order 109
Minimal Bounds w.r.t. Stochastic, Hazard, and Likelihood Ratio Order 111
Lorenz Order and Convex Orders 111
3.2.2 Multivariate Stochastic Orderings and Its Applications to Progressively Type-II Censored Order Statistics 113
Stochastic Order 113
Likelihood Ratio Order 113
3.2.3 Applications to Spacings of Progressively Type-II Censored Order Statistics 116
3.3 Aging Properties 118
3.4 Asymptotic and Extreme Value Results 119
3.4.1 Extreme Value Analysis for Order Statistics 119
3.4.2 Extreme Value Analysis for Progressively Type-II Censored Order Statistics 120
3.4.3 Extreme Value Analysis for Exponential Progressively Type-II Censored order statistics 123
3.4.4 Extreme Value Analysis for Progressively Type-II Censored Order Statistics from a Cumulative Distribution Function F 124
Positive Asymptotic Variance 124
Zero Asymptotic Variance 125
3.4.5 Applications to Upper, Lower, Central, and Intermediate Progressively Type-II Censored Order Statistics 127
Upper Progressively Type-II Censored Order Statistics 128
Lower Progressively Type-II Censored Order Statistics 128
Central and Intermediate Progressively Type-II Censored Order Statistics 129
3.4.6 Limits for Central Progressively Type-II Censored Order Statistics with Blocked Observations 130
3.5 Near Minimum Progressively Type-II CensoredOrder Statistics 133
Chapter4 Progressive Type-I Censoring: Basic Properties 135
4.1 Distribution and Block Independence 135
4.2 Number of Observations 142
Chapter5 Progressive Hybrid Censoring: Distributions and Properties 145
5.1 Type-I Progressive Hybrid Censoring 145
5.1.1 Spacings for Exponential Distribution 147
5.1.2 Distributions of Total Time on Test and Related Statistics 150
5.1.3 Moment Generating Function 152
5.2 Type-II Progressive Hybrid Censoring 156
5.2.1 Exponential Distributions 157
5.3 Generalized Progressive Hybrid Censoring 160
Generalized Progressive Type-I Hybrid Censoring 161
Generalized Progressive Type-II Hybrid Censoring 161
Chapter6 Adaptive Progressive Type-II Censoring and Related Models 163
6.1 General Model of Adaptive Progressive Type-II Censoring 164
Distributional Assumptions 165
6.2 Particular Models 170
6.2.1 Nonadaptive Type-II Progressive Censoring 170
6.2.2 Ng–Kundu–Chan Model 170
6.2.3 Flexible Progressive Censoring 171
6.2.4 Progressive Censoring with Random Removals 172
Chapter7 Moments of Progressively Type-II Censored Order Statistics 174
7.1 General Distributions 174
7.1.1 Representations for Moments 174
7.1.2 Existence of Moments 175
7.2 Moments for Particular Distributions 180
7.2.1 Exponential Distribution 180
7.2.2 Weibull Distributions 181
7.2.3 Reflected Power Distribution 182
7.2.4 Uniform Distribution 183
7.2.5 Pareto Distribution 184
7.2.6 Lomax Distribution 185
7.2.7 Extreme Value Distribution 185
7.3 Recurrence Relations for Moments 186
7.3.1 General Results 186
7.3.2 Results for Particular Distributions 186
Exponential Distribution 187
Truncated Exponential Distribution 189
Truncated Pareto Distributions 191
Truncated Power Distributions 193
Truncated Reflected Power Distributions 194
Logistic and Related Distributions 196
Doubly Truncated Burr Distributions 197
7.4 Moments for Symmetric Distributions 198
7.5 Bounds for Moments 201
7.5.1 Bounds Based on the Cauchy–Schwarz Inequality 201
7.5.2 Bound Based on the Method of Greatest Convex Minorant 203
7.5.3 Further Bounds 206
7.6 First-Order Approximations to Moments 209
Chapter8 Simulation of Progressively Censored Order Statistics 211
8.1 Generation of Progressively Type-II Censored Order Statistics 211
8.1.1 Generation of General Progressively Type-II Censored Order Statistics 214
8.1.2 Generation of Progressively Type-II Censored Order Statistics from a One-Step Censoring Plan 215
8.1.3 Simulation of Progressively Hybrid Censored Data 215
8.2 Progressively Type-I Censored Data 216
8.3 Progressively Type-I Interval Censored Data 217
Chapter9 Information Measures 219
9.1 Fisher Information in Progressively Type-II CensoredSamples 219
9.1.1 Hazard Rate Representation of Fisher Information 219
Fisher Information in Location or Scale Family 221
Fisher Information in the Multiparameter Case 222
9.1.2 Fisher Information via Missing Information Principle 224
9.1.3 Fisher Information for Particular Distributions 227
Invariance of Fisher Information Under Progressive Censoring 227
Weibull Distribution (Shape) and Extreme Value Distribution (Scale) 227
Laplace Distribution (Location) 229
Logistic Distribution 229
Normal Distribution 230
Lomax Distribution 230
9.1.4 Recurrence Relations for Fisher Information 230
9.2 Fisher Information in Progressive Hybrid Censoring 232
9.3 Tukey's Linear Sensitivity Measure 233
9.4 Entropy 234
9.5 Kullback–Leibler Information 239
9.6 Pitman Closeness 242
Chapter10 Progressive Type-II Censoring Under Nonstandard Conditions 247
10.1 Mixture Representation for Progressively Type-II Censored Order Statistics with Arbitrary Distribution 247
10.2 Joint Density Function of Progressively Type-II Censored Order Statistics 249
Modeling of Outliers 252
Connection to Permanents 253
Applications to Stochastic Orderings 254
10.3 Dependence Structure of INID Progressively Type-II Censored Order Statistics 257
10.4 Dependence and Copulas 259
10.5 Progressive Type-II Censoring for Multivariate Observations 261
Part II Inference 263
Chapter11 Linear Estimation in Progressive Type-II Censoring 264
11.1 Preliminaries 264
11.1.1 Least-Squares Estimation 265
11.1.2 Linear Equivariant Estimation 267
11.1.3 First-Order Approximations to BLUEs and BLEEs 268
11.2 Linear Estimation for Particular Distributions 269
11.2.1 Exponential Distributions 269
Linear Estimates 270
11.2.2 Generalized Pareto Distributions 272
Uniform Distribution 276
Pareto Distribution 277
Lomax Distribution 279
11.2.3 Weibull and Extreme Value Distributions 279
11.2.4 Laplace Distribution 280
11.2.5 Logistic Distributions 282
11.3 Asymptotic Best Linear Unbiased Estimatorsfor Blocked Progressively Type-II Censored Order Statistics 282
Chapter12 Maximum Likelihood Estimation in Progressive Type-IICensoring 284
12.1 Exponential Distribution 285
Scale Parameter Unknown 285
Location Parameter Unknown 287
Location and Scale Parameters Unknown 287
12.2 Weibull Distribution 291
12.3 Reflected Power Distribution 294
Shape Parameter Known 294
Shape Parameter Unknown 296
12.4 Uniform Distribution 297
Location Parameter ? Known 297
Location Parameter ? Unknown 298
12.5 Pareto Distributions 298
Shape Parameter Unknown, Scale Parameter Known 299
Shape Parameter Known, Scale Parameter Unknown 300
Shape and Scale Parameter Unknown 301
Shape and Scale Parameter Unknown with Equal Location Scale 302
12.6 Laplace Distribution 303
Location Parameter ?R Known and Scale Parameter > 0 Unknown
Location Parameter ?R Unknown and Scale Parameter > 0 Known
Location Parameter ?R and Scale Parameter > 0 Unknown
12.7 Some Other Location–Scale Families 312
12.7.1 Weibull Distributions 313
12.7.2 Normal Distributions 314
12.7.3 Log-Normal Distributions 316
12.7.4 Extreme Value Distribution (Type I) 317
12.7.5 Logistic Distribution 317
12.8 Other Distributions 318
12.9 Related Methods 320
12.9.1 Modified Maximum Likelihood Estimation 321
12.9.2 Approximate Maximum Likelihood Estimation 322
Extreme Value Distribution 322
Weibull Distribution 323
Other Distributions 324
12.10 M-Estimation 325
12.11 Order Restricted Inference 327
Chapter13 Point Estimation in Progressive Type-I Censoring 330
13.1 Exponential Distribution 331
13.1.1 One-Parameter Exponential Distribution 331
13.1.2 Two-Parameter Exponential Distribution 334
13.1.3 Modified Moment Estimation 336
13.2 Weibull Distributions 337
13.3 Extreme Value Distributions 339
13.4 Normal Distribution 339
13.5 Burr-XII Distribution 341
13.6 Logistic Distributions 342
Chapter14 Progressive Hybrid and Adaptive Censoring and RelatedInference 343
14.1 Likelihood Inference for Type-I Progressive Hybrid Censored Data 344
14.1.1 Likelihood Inference for Two-Parameter Exponential Distributions 344
Location Parameter Known 344
Confidence Intervals 345
Location Parameter Unknown 348
14.1.2 Other Distributions 350
14.2 Likelihood Inference for Type-II Progressive Hybrid Censored Data 351
14.2.1 Exponential Distribution 351
14.2.2 Other Distributions 352
14.3 Inferential Results for Adaptive Progressive Type-IICensoring 352
14.3.1 Ng–Kundu–Chan Model 354
14.3.2 Progressive Censoring with Random Removals 355
Chapter15 Bayesian Inference for Progressively Type-II Censored Data 357
15.1 Exponential and Weibull Distributions 358
15.2 Rayleigh Distribution 363
15.3 Pareto Distribution 365
15.4 Burr Distributions 367
15.5 Other Distributions 368
Chapter16 Point Prediction from Progressively Type-II Censored Samples 370
16.1 Prediction Concepts 370
16.2 Prediction of Failure Times of Censored Units 372
16.2.1 Exponential Distribution 373
Scale Parameter > 0 Known
Scale Parameter > 0 Unknown
16.2.2 Extreme Value Distribution 378
Scale Parameter > 0 Known
Scale Parameter > 0 Unknown
16.2.3 Normal Distribution 381
Location Parameter ? Known 381
Location Parameter ? Unknown 381
16.2.4 Pareto Distributions 384
16.3 Prediction of Future Observations 387
16.3.1 Linear Prediction 387
16.3.2 Bayesian Prediction 388
Bayesian Prediction: One-Sample Case 389
Bayesian Prediction: Two-Sample Case 390
Chapter17 Statistical Intervals for Progressively Type-II Censored Data 393
17.1 Exact Confidence Intervals 393
17.1.1 Exponential Distribution 393
17.1.2 Weibull Distribution 398
17.1.3 Pareto Distribution 402
17.1.4 Other Parametric Distributions 405
17.1.5 Nonparametric Confidence Intervals for Quantiles 406
17.1.6 An Excursus: Two-Sample Nonparametric Confidence Intervals from Type-II Censored Data 412
17.2 Conditional Statistical Intervals 416
17.2.1 Inference in a General Location–Scale Family 416
Conditional Confidence Intervals for Location and Scale Parameters 417
Conditional Confidence Intervals for Quantiles 418
Conditional Confidence Intervals for Reliability 419
Conditional Prediction Intervals for Future Failure Times 419
17.2.2 Exponential Distribution 420
17.2.3 Extreme Value Distribution 420
17.2.4 Log-Gamma Distribution 423
17.2.5 Pareto Distribution 424
17.2.6 Laplace Distribution 425
17.2.7 Other Distributions 428
17.3 Asymptotic Confidence Intervals 428
17.4 Prediction Intervals 429
17.4.1 Nonparametric Prediction Intervals 429
17.4.2 Parametric and Bayesian Prediction 429
Prediction Intervals for Censored Failure Times 430
Prediction Intervals for Future Observations in the Same Sample 430
Prediction Intervals for Observations of an Independent Future Sample from the Same Population 431
17.5 Nonparametric Tolerance Intervals 432
17.6 Highest Posterior Density Credible Intervals 432
Chapter18 Progressive Type-I Interval Censored Data 434
18.1 Parametric Inference 434
18.2 Optimal Inspection Times 437
18.3 Optimal Progressive Interval Censoring Proportions 438
Chapter19 Goodness-of-Fit Tests in Progressive Type-II Censoring 439
19.1 Tests on Exponentiality 439
19.2 Goodness-of-Fit Tests for Other Distributional Assumptions 443
19.2.1 Methods Based on Spacings and Deviation from the Uniform Distribution 443
19.2.2 Tests Based on Empirical Distribution Function 446
19.2.3 Tests Based on Kullback–Leibler Distance 448
Chapter20 Counting and Quantile Processes and Progressive Censoring 450
20.1 Counting Process Approach 450
20.1.1 Semiparametric Proportional Hazards Model 458
20.2 Quantile Process Approach 459
Chapter21 Nonparametric Inferential Issues in Progressive Type-IICensoring 462
21.1 Precedence-Type Nonparametric Tests 462
21.1.1 Precedence-Type Nonparametric Tests with Progressive Censoring 463
Distribution of Test Statistics 465
Precedence-Type Test Based on Kaplan–Meier Estimator of Cumulative Distribution Function 467
Two Progressively Censored Samples 469
21.1.2 Tests for Hazard Rate Ordering 472
Part III Applications in Survival Analysis and Reliability 476
Chapter22 Acceptance Sampling Plans 477
22.1 Exponential Distribution 478
22.1.1 Acceptance Sampling Plans Without Consumer Risk 478
One-Sided Sampling Plans 479
Two-Sided Sampling Plans 481
22.1.2 Acceptance Sampling Plans with Consumer Risk 482
22.1.3 Bayesian Variable Sampling Plans with Progressive Hybrid Censoring 484
22.2 Weibull Distribution 485
22.3 Log-Normal Distribution 486
22.4 Reliability Sampling Plans for Interval Censored Data 486
22.5 Capability Indices 487
22.5.1 Exponential Progressively Type-II Censored Order Statistics 487
22.5.2 Other Distributions 490
Chapter23 Accelerated Life Testing 491
23.1 Step-Stress Models 491
23.1.1 Inference for Simple Step-Stress Model Under Progressive Type-II Censoring 493
Progressively Type-II Censored Step-Stress Data 493
Likelihood Function and Maximum Likelihood Estimation 495
Exact Conditional Distributions of MLEs 496
Confidence Intervals for 1 and 2 499
Optimal Censoring and Optimal Test Plan 500
23.1.2 Inference for a Simple Step-Stress Model with Random Change Under Progressive Type-II Censoring 500
Connection to Sequential Order Statistics 502
Parametrization via Log-Linear Link Function 503
23.1.3 Inference for Multiple Step-Stress Model Under Progressive Type-I Censoring 505
Progressively Type-I Censored Step-Stress Data 505
Likelihood Function and MLEs 505
Optimal Step-Stress Test 508
A Modified Progressive Censoring Scheme and Optimal Step-Stress Test 509
Step-Stress Test with Link Function Based on Box–Cox Transformation 510
Progressively Type-I Interval Censored Exponential Data 511
23.1.4 Multiple Step-Stress Model with Progressive Censoring: An Approach Based on Sequential Order Statistics 513
23.2 Progressive Stress Models 513
Chapter24 Stress–Strength Models with Progressively Censored Data 516
24.1 Exponentially Distributed Stress and Strength 517
24.1.1 Exponentially Distributed Stress and Strength with Known Location Parameter 517
24.1.2 Exponentially Distributed Stress and Strength with Common Unknown LocationParameter 518
24.2 Further Stress–Strength Distributions 521
Chapter25 Multi-sample Models 523
25.1 Competing Risk Models 523
25.1.1 Model and Notation 523
25.1.2 Exponential Distribution 524
25.1.3 Weibull Distributions 528
25.1.4 Lomax Distribution 530
25.2 Joint Progressive Censoring 531
25.3 Concomitants 533
25.3.1 Missing Information Principle and EM-Algorithm 535
25.4 Progressively Censored Systems Data 536
25.4.1 Progressive First-Failure Censoring: Series Systems 537
25.4.2 Parallel Systems 537
Chapter26 Optimal Experimental Designs 539
26.1 Preliminaries 540
26.2 Probabilistic Criteria 542
26.3 Precision of Estimates 544
26.3.1 Exponential Distribution 546
26.3.2 Generalized Pareto Distributions 546
26.3.3 Extreme Value Distribution 550
26.3.4 Further Distributions 550
26.4 Maximum Fisher Information 552
26.4.1 Single Parameter Case 552
Optimal One-Step Plans 554
State of the Art 558
26.4.2 Two-Parameter Case 559
26.4.3 Asymptotically Optimal Censoring Schemes 565
26.4.4 Maximum Fisher Information Plans in Progressive Hybrid Censoring 567
26.5 Other Optimality Criteria and Approaches 567
26.5.1 Maximum Entropy Plans 567
26.5.2 Optimal Estimation of Quantiles 568
26.5.3 Optimization Based on Pitman Closeness 570
26.5.4 Optimal Block Censoring 573
Computational Results 576
26.5.5 Other Criteria for Optimal Censoring Plans 578
Appendix A Distributions 579
A.1 Definitions of Distributions 579
A.2 Definitions and Preliminaries 582
A.2.1 Quantile Function 582
A.2.2 Stochastic Orders 583
Univariate Stochastic Orders 583
A.2.2.1 Multivariate Stochastic Orderings 585
Orderings of Real Vectors 586
Appendix B Additional Demonstrative Data Sets 587
B.1 Progressively Type-II Censored Data 587
B.2 Progressively Type-I Censored Data 589
Notation 593
References 598
Author Index 636
Index 646
| Erscheint lt. Verlag | 24.7.2014 |
|---|---|
| Reihe/Serie | Statistics for Industry and Technology | Statistics for Industry and Technology |
| Zusatzinfo | XXI, 645 p. 47 illus., 4 illus. in color. |
| Verlagsort | New York |
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Angewandte Mathematik |
| Mathematik / Informatik ► Mathematik ► Statistik | |
| Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik | |
| Medizin / Pharmazie ► Allgemeines / Lexika | |
| Technik | |
| Schlagworte | Experimental Design • Fisher Information • Goodness-of-fit test • life testing • nonparametric inference • Optimal progressive censoring • Ordered Data • Parametric inference • Progressive censoring • Progressively censored samples • quality control • Reliability |
| ISBN-10 | 0-8176-4807-0 / 0817648070 |
| ISBN-13 | 978-0-8176-4807-7 / 9780817648077 |
| Haben Sie eine Frage zum Produkt? |
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