The Weibull Distribution - gbv.de
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The Weibull Distribution A Handbook Horst Rinne Justus-Liebig-University Giessen, Germany @ CRC Press Taylor &. Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group an informa business A CHAPMAN & HALL BOOK
Transcript of The Weibull Distribution - gbv.de
The Weibull
Distribution A Handbook
Horst Rinne Justus-Liebig-University
Giessen, Germany
@ CRC Press Taylor &. Francis Group
Boca Raton London New York
CRC Press is an imprint of the Taylor & Francis Group an informa business
A CHAPMAN & HALL BOOK
Contents
Preface XIII
List of Figures XVII
List of Tables XXI
I Genesis, theory and description 1
1 History and meaning of the WEIBULL distribution 3
1.1 Genesis of the WEIBULL distribution 3
1.1.1 Origins in science 4
1.1.2 Origins in practice 9
1.1.2.1 Grinding of material — ROSIN, RAMMLER and SPERLING 9
1.1.2.2 Strength of material — WEIBULL 12
1.2 Physical meanings and interpretations of the WEIBULL distribution . . . . 15
1.2.1 The model of the weakest link 15
1.2.2 Two models of data degradation leading to WEIBULL distributed failures 19
1.2.3 The hazard rate approach 22
1.2.4 The broken-stick model 24
2 Definition and properties of the WEIBULL distribution 27
2.1 Functions describing lifetime as a random variable 27
2.2 Failure density 30
2.2.1 Three-parameter density 30
2.2.2 Two- and one-parameter densities 34
2.2.3 Analysis of the reduced WEIBULL density 36
2.2.4 Differing notations 41
2.3 Failure distribution (CDF) and reliability function (CCDF) 43
2.4 Hazard rate (HR) 46
2.5 Cumulative hazard rate (CHR) 49
2.6 Mean residual life function (MRL) 51
IV Contents
2.7 Aging criteria 58
2.8 Percentiles and random number generation 68
2.8.1 Percentiles 68
2.8.2 WEIBULL random numbers 70
2.9 Moments, cumulants and their generating functions 71
2.9.1 General formulas 71
2.9.2 Mean and its relation to mode and median 85
2.9.3 Variance, standard deviation and coefficient of variation 89
2.9.4 Skewness and kurtosis 91
3 Related distributions 98
3.1 Systems of distributions and the WEIBULL distribution 98
3.1.1 PEARSON system 98
3.1.2 BURR system 101
3.1.3 JOHNSON system 103
3.1.4 Miscellaneous 105
3.2 WEIBULL distributions and other familiar distributions 108
3.2.1 WEIBULL and exponential distributions 108
3.2.2 WEIBULL and extreme value distributions 108
3.2.3 WEIBULL and gamma distributions I l l
3.2.4 WEIBULL and normal distributions 112
3.2.5 WEIBULL and further distributions 115
3.3 Modifications of the WEIBULL distribution 119
3.3.1 Discrete WEIBULL distribution 119
3.3.2 Reflected and double WEIBULL distributions 125
3.3.3 Inverse WEIBULL distribution 129
3.3.4 Log-WEIBULL distribution 131
3.3.5 Truncated WEIBULL distributions 133
3.3.6 Models including two or more distributions 137
3.3.6.1 WEIBULL folding 138
3.3.6.2 WEIBULL models for parallel and series systems 141
3.3.6.3 Composite WEIBULL distributions 146
3.3.6.4 Mixed WEIBULL distributions 149
3.3.6.5 Compound WEIBULL distributions 155
Contents V
3.3.7 WEIBULL distributions with additional parameters 158
3.3.7.1 Four-parameter distributions 158
3.3.7.2 Five-parameter distributions 166
3.3.8 WEIBULL distributions with varying parameters 168
3.3.8.1 Time-dependent parameters 168
3.3.8.2 Models with covariates 170
3.3.9 Multidimensional WEIBULL models 173
3.3.9.1 Bivariate WEIBULL distributions 173
3.3.9.2 Multivariate WEIBULL distributions 184
3.3.10 Miscellaneous 186
4 WEIBULL processes and WEIBULL renewal theory 189
4.1 Stochastic processes—An overview 189
4.2 POISSON processes 193
4.3 WEIBULL processes 199
4.4 WEIBULL renewal processes 202
4.4.1 Renewal processes 202
4.4.2 Ordinary WEIBULL renewal process 213
4.4.2.1 Time to the n-th renewal 214
4.4.2.2 Number of renewals Nt 216
4.4.2.3 Forward and backward recurrence times 220
5 Order statistics and related variables 223
5.1 General definitions and basic formulas 223
5.1.1 Distributions and moments of order statistics 223
5.1.2 Functions of order statistics 228
5.1.3 Record times and record values 231
5.2 WEIBULL order statistics 237
5.3 WEIBULL record values 244
5.4 Log-WEIBULL order statistics 246
5.5 Order statistics and record values for several related WEIBULL distributions 250
6 Characterizations 254
6.1 WEIBULL characterizations based on functional equations 254
VI Contents
6.2 WEIBULL characterizations based on conditional moments 259
6.3 WEIBULL characterizations based on order statistics 264
6.4 Miscellaneous approaches of WEIBULL characterizations 268
6.5 Characterizations of related WEIBULL distributions 271
II Applications and inference 273
7 WEIBULL applications and aids in doing WEIBULL analysis 275
7.1 A survey of WEIBULL applications 275
7.2 Aids in WEIBULL analysis 285
8 Collecting life data 286
8.1 Field data versus laboratory data 286
8.2 Parameters of a life test plan 287
8.3 Types of life test plans 290
8.3.1 Introductory remarks 291
8.3.2 Singly censored tests 291
8.3.2.1 Type-I censoring 292
8.3.2.2 Type-II censoring 296
8.3.2.3 Combined type-I and type-II censoring 301
8.3.2.4 Indirect censoring 302
8.3.3 Multiply censored tests 305
8.3.4 Further types of censoring 310
9 Parameter estimation — Graphical approaches 313
9.1 General remarks on parameter estimation 313
9.2 Motivation for and types of graphs in statistics 317
9.2.1 PP-plots and QQ-plots 317
9.2.2 Probability plots 322
9.2.2.1 Theory and construction 322
9.2.2.2 Plotting positions 326
9.2.2.3 Advantages and limitations 329
9.2.3 Hazard plot 331
9.2.4 TTT-plot 333
Contents VII
9.3 WEIBULL plotting techniques 335
9.3.1 Complete samples and singly censored samples 336
9.3.2 Multiply censored data 342
9.3.2.1 Probability plotting 342
9.3.2.2 Hazard Plotting 346
9.3.3 Special problems 347
9.3.3.1 Three-parameter WEIBULL distribution 347
9.3.3.2 Mixed WEIBULL distributions 352
9.4 Nomograms and supporting graphs 354
10 Parameter estimation — Least squares and linear approaches 355
10.1 From OLS to linear estimation 356
10.2 BLUEs for the Log-WEIBULL distribution 359
10.2.1 Complete and singly type-II censored samples 360
10.2.2 Progressively type-II censored samples 364
10.3 BLIEs for Log-WEIBULL parameters 368
10.3.1 BLUE versus BLIE 368
10.3.2 Type-II censored samples 371
10.3.3 Type-I censored samples 374
10.4 Approximations to BLUEs and BLIEs 375
10.4.1 Least squares with various functions of the variable 375
10.4.2 Linear estimation with linear and polynomial coefficients 377
10.4.3 GLUEs of BAIN and ENGELHARDT 382
10.4.4 BLOM's unbiased nearly best linear estimator 383
10.4.5 ABLIEs 384
10.5 Linear estimation with a few optimally chosen order statistics 387
10.5.1 Optimum-order statistics for small sample sizes 387
10.5.2 Quantile estimators and ABLEs 391
10.6 Linear estimation of a subset of parameters 394
10.6.1 Estimation of one of the Log-WEIBULL parameters 394
10.6.2 Estimation of one or two of the three WEIBULL parameters . . . . 395
10.6.2.1 Estimating a and b with с known 396
10.6.2.2 Estimating either b or с 397
10.7 Miscellaneous problems of linear estimation 399
Contents
11 Parameter estimation — Maximum likelihood approaches 402 11.1 Likelihood functions and likelihood equations 402
11.2 Statistical and computational aspects of MLEs 405
11.2.1 Asymptotic properties of MLEs 406
11.2.2 Iterated MLEs 413
11.3 Uncensored samples with non-grouped data 417
11.3.1 Two-parameter WEIBULL distribution 417
11.3.1.1 Point and interval estimates for b and с 417
11.3.1.2 Finite sample results based on pivotal functions 421
11.3.2 Three-parameter WEIBULL distribution 426
11.3.2.1 History of optimizing the WEIBULL log-likelihood . . .426
11.3.2.2 A non-failing algorithm 428
11.3.2.3 Modified ML estimation 430
11.3.2.4 Finite sample results 433
11.4 Uncensored samples with grouped data 434
11.5 Samples censored on both sides 436
11.6 Samples singly censored on the right 438
11.6.1 Two-parameter WEIBULL distribution 438
11.6.1.1 Solving the likelihood equations 438
11.6.1.2 Statistical properties of the estimators 442
11.6.2 Three-parameter WEIBULL distribution 446
11.7 Samples progressively censored on the right 448
11.7.1 Type-I censoring 449
11.7.2 Type-II censoring 450
11.8 Randomly censored samples 453
12 Parameter estimation — Methods of moments 455 12.1 Traditional method of moments 455
12.1.1 Two-parameter WEIBULL distribution 456
12.1.2 Three-parameter WEIBULL distribution 464
12.2 Modified method of moments 467
12.3 W. WEIBULL'S approaches to estimation by moments 470
12.4 Method of probability weighted moments 473
12.5 Method of fractional moments 474
Contents IX
13 Parameter estimation — More classical approaches and comparisons 476
13.1 Method of percentiles 476
13.1.1 Two-parameter WEIBULL distribution 476
13.1.2 Three-parameter WEIBULL distribution 480
13.2 Minimum distance estimators 485
13.3 Some hybrid estimation methods 488
13.4 Miscellaneous approaches 491
13.4.1 MENON'S estimators 491
13.4.2 Block estimators of HÜSLER/SCHÜPBACH 493
13.4.3 KAPPENMAN'S estimators based on the likelihood ratio 494
13.4.4 KAPPENMAN'S estimators based on sample reuse 495
13.4.5 Confidence intervals for b and с based on the quantiles of beta distributions 496
13.4.6 Robust estimation 497
13.4.7 Bootstrapping 498
13.5 Further estimators for only one of the WEIBULL parameters 498
13.5.1 Location parameter 498
13.5.2 Scale parameter 501
13.5.3 Shape parameter 503
13.6 Comparisons of classical estimators 508
14 Parameter estimation — BAYESIAN approaches 511
14.1 Foundations of BAYESIAN inference 511
14.1.1 Types of distributions encountered 511
14.1.2 BAYESIAN estimation theory 513
14.1.3 Prior distributions 515
14.2 Two-parameter WEIBULL distribution 517
14.2.1 Random scale parameter and known shape parameter 517
14.2.2 Random shape parameter and known scale parameter 525
14.2.3 Random scale and random shape parameters 526
14.3 Empirical BAYES estimation 528
15 Parameter estimation — Further approaches 531
15.1 Fiducial inference 531
Contents
15.1.1 The key ideas 531
15.1.2 Application to the WEIBULL parameters 532
15.2 Structural inference 532
15.2.1 The key ideas 533
15.2.2 Application to the WEIBULL parameters 534
16 Parameter estimation in accelerated life testing 536
16.1 Life-stress relationships 536
16.2 ALT using constant stress models 541
16.2.1 Direct ML procedure of the IPL-WEIBULL model 543
16.2.2 Direct ML estimation of an exponential life-stress relationship . . . 544
16.2.3 MLE of a log-linear life-stress relationship 546
16.3 ALT using step-stress models 548
16.4 ALT using progressive stress models 551
16.5 Models for PALT 553
17 Parameter estimation for mixed WEIBULL models 557
17.1 Classical estimation approaches 558
17.1.1 Estimation by the method of moments 558
17.1.2 Estimation by maximum likelihood 559
17.1.2.1 The case of two subpopulations 559
17.1.2.2 The case of m subpopulations (m > 2) 561
17.2 BAYESIAN estimation approaches 567
17.2.1 The case of two subpopulations 567
17.2.2 The case of m subpopulations (m > 2) 570
18 Inference of WEIBULL processes 571
18.1 Failure truncation 572
18.1.1 The case of one observed process 572
18.1.2 The case of more than one observed process 577
18.2 Time truncation 579
18.2.1 The case of one observed process 579
18.2.2 The case of more than one observed process 580
18.3 Other methods of collecting data 581
18.4 Estimation based on DUANE'S plot 582
Contents XI
19 Estimation of percentiles and reliability including tolerance limits 585
19.1 Percentiles, reliability and tolerance intervals 585
19.2 Classical methods of estimating reliability R(x) 588
19.2.1 Non-parametric approaches 589
19.2.2 Maximum likelihood approaches 591
19.3 Classical methods of estimating percentiles xp 596
19.3.1 Anon-parametric approach 597
19.3.2 Maximum likelihood approaches 597
19.4 Tolerance intervals 600
19.4.1 Anon-parametric approach 600
19.4.2 Maximum likelihood approaches 601
19.5 BAYESIAN approaches 606
20 Prediction of future random quantities 610
20.1 Classical prediction 610
20.1.1 Prediction for a WEIBULL process 610
20.1.2 One-sample prediction 613
20.1.3 Two-sample prediction 616
20.1.4 Prediction of failure numbers 619
20.2 BAYESIAN prediction 622
21 WEIBULL parameter testing 624
21.1 Testing hypotheses on function parameters 624
21.1.1 Hypotheses concerning the shape parameter с 624
21.1.1.1 Tests for one parameter 625
21.1.1.2 Tests for к > 2 parameters 631
21.1.2 Hypotheses concerning the scale parameter b 634
21.1.3 Hypotheses concerning the location parameter a 640
21.1.4 Hypotheses concerning two or more parameters 644
21.2 Testing hypotheses on functional parameters 646
21.2.1 Hypotheses concerning the mean ß 646
21.2.2 Hypotheses concerning the variance a2 647
21.2.3 Hypotheses on the reliability R(x) 648
21.2.4 Hypotheses concerning the percentile xp 649
XII Contents
22 WEIBULL goodness-of-fit testing and related problems 651
22.1 Goodness-of-fit testing 651
22.1.1 Tests of x2-type 652
22.1.2 Tests based on EDF statistics 652
22.1.2.1 Introduction 653
22.1.2.2 Fully specified distribution and uncensored sample . . . .655
22.1.2.3 Fully specified distribution and censored sample 657
22.1.2.4 Testing a composite hypothesis 663
22.1.3 Tests using other than EDF statistics 667
22.1.3.1 Tests based on the ratio of two estimates of scale 667
22.1.3.2 Tests based on spacings and leaps 669
22.1.3.3 Correlation tests 672
22.2 Discrimination between WEIBULL and other distributions 674
22.2.1 Discrimination between the two-parameter and three-parameter WEIBULL distributions 674
22.2.2 Discrimination between WEIBULL and one other distribution . . . . 676
22.2.2.1 WEIBULL versus exponential distribution 676
22.2.2.2 WEIBULL versus gamma distribution 679
22.2.2.3 WEIBULL versus lognormal distribution 681
22.2.3 Discrimination between WEIBULL and more than one other distribution 685
22.3 Selecting the better of several WEIBULL distributions 686
III Appendices 691
Table of the gamma, digamma and trigamma functions 693
Abbreviations 695
Mathematical and statistical notations 698
Bibliography 701
Author index 763
Subject index 777
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