The Weibull distribution: a handbook
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Boca Raton, Fla. [u.a.]
CRC Press
2009
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIV, 784 S. graph. Darst. |
| ISBN: | 9781420087437 |
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| adam_text | 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.................HI
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 »-th renewal ................. 214
4.4.2.2 Number of renewals Xf.................. 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-WElBULL 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 GLUEsof 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 c known.............396
10.6.2.2 Estimating either b or c..................397
10.7 Miscellaneous problems of linear estimation.................399
VIII______________________________________________________________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 c.......... 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 HUSLER/SCHUPBACH.............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 c 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
X 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 forPALT...............................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 /?(. ¦)............... 588
19.2.1 Non-parametric approaches ..................... 589
19.2.2 Maximum likelihood approaches................... 591
19.3 Classical methods of estimating percentiles .;¦/ ............... 596
19.3.1 Anon-parametric approach...................... 597
19.3.2 Maximum likelihood approaches................... 597
19.4 Tolerance intervals .............................. 600
19.4.1 A non-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 Weibuli. 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 c............624
21.1.1.1 Tests for one parameter..................625
21.1.1.2 Tests for k 2 parameters ................631
21.1.2 Hypotheses concerning the scale parameter/ ............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// .................646
21.2.2 Hypotheses concerning the variance a2...............647
21.2.3 Hypotheses on the reliability/?(.;)..................648
21.2.4 Hypotheses concerning the percentile .rp..............649
XII Contents
22 Weibull goodness-of-iit 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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| indexdate | 2024-07-09T21:40:27Z |
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| isbn | 9781420087437 |
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| spelling | Rinne, Horst 1939-2023 Verfasser (DE-588)115833129 aut The Weibull distribution a handbook Horst Rinne Boca Raton, Fla. [u.a.] CRC Press 2009 XXIV, 784 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Weibull-Verteilung (DE-588)4065029-7 gnd rswk-swf Weibull-Verteilung (DE-588)4065029-7 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017616557&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rinne, Horst 1939-2023 The Weibull distribution a handbook Weibull-Verteilung (DE-588)4065029-7 gnd |
| subject_GND | (DE-588)4065029-7 |
| title | The Weibull distribution a handbook |
| title_auth | The Weibull distribution a handbook |
| title_exact_search | The Weibull distribution a handbook |
| title_full | The Weibull distribution a handbook Horst Rinne |
| title_fullStr | The Weibull distribution a handbook Horst Rinne |
| title_full_unstemmed | The Weibull distribution a handbook Horst Rinne |
| title_short | The Weibull distribution |
| title_sort | the weibull distribution a handbook |
| title_sub | a handbook |
| topic | Weibull-Verteilung (DE-588)4065029-7 gnd |
| topic_facet | Weibull-Verteilung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017616557&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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