Using the Weibull Distribution: Reliability, Modeling, and Inference

364 Pages·2012·2.514 MB·English

by  John I. McCool(auth.)

Using the Weibull Distribution
WILEY SERIES IN PROBABILITY AND STATISTICS
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A complete list of the titles in this series appears at the end of this volume.
Using the Weibull Distribution
Reliability, Modeling, and Inference
JOHN I. McCOOL
A JOHN WILEY & SONS, INC., PUBLICATION
Cover image: Courtesy of the author.
Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved.
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Library of Congress Cataloging-in-Publication Data:
McCool, John, 1936–
Using the Weibull distribution : reliability, modeling, and inference / John I. McCool.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-118-21798-6 (cloth)
1. Weibull distribution—Textbooks. 2. Probabilities—Textbooks. I. Title.
QA273.6.M38 2012
519.2'4–dc23
2012002909
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
To Jim, Fran, and Peg
and
To Mary, John, and Jen
Contents
Preface xiii
1. Probability 1
1.1 Sample Spaces and Events, 2
1.2 Mutually Exclusive Events, 2
1.3 Venn Diagrams, 3
1.4 Unions of Events and Joint Probability, 4
1.5 Conditional Probability, 6
1.6 Independence, 8
1.7 Partitions and the Law of Total Probability, 9
1.8 Reliability, 12
1.9 Series Systems, 12
1.10 Parallel Systems, 13
1.11 Complex Systems, 15
1.12 Crosslinked Systems, 16
1.13 Reliability Importance, 19
References, 20
Exercises, 21
2. Discrete and Continuous Random Variables 23
2.1 Probability Distributions, 24
2.2 Functions of a Random Variable, 26
2.3 Jointly Distributed Discrete Random Variables, 28
2.4 Conditional Expectation, 32
2.5 The Binomial Distribution, 34
2.5.1 Confi dence Limits for the Binomial Proportion p, 38
2.6 The Poisson Distribution, 39
vii
viii contents
2.7 The Geometric Distribution, 41
2.8 Continuous Random Variables, 42
2.8.1 The Hazard Function, 49
2.9 Jointly Distributed Continuous Random Variables, 51
2.10 Simulating Samples from Continuous Distributions, 52
2.11 The Normal Distribution, 54
2.12 Distribution of the Sample Mean, 60
2.12.1 P[X<Y] for Normal Variables, 65
2.13 The Lognormal Distribution, 66
2.14 Simple Linear Regression, 67
References, 69
Exercises, 69
3. Properties of the Weibull Distribution 73
3.1 The Weibull Cumulative Distribution Function (CDF),
Percentiles, Moments, and Hazard Function, 73
3.1.1 Hazard Function, 75
3.1.2 The Mode, 77
3.1.3 Quantiles, 77
3.1.4 Moments, 78
3.2 The Minima of Weibull Samples, 82
3.3 Transformations, 83
3.3.1 The Power Transformation, 83
3.3.2 The Logarithmic Transformation, 84
3.4 The Conditional Weibull Distribution, 86
3.5 Quantiles for Order Statistics of a Weibull Sample, 89
3.5.1 The Weakest Link Phenomenon, 92
3.6 Simulating Weibull Samples, 92
References, 94
Exercises, 95
4. Weibull Probability Models 97
4.1 System Reliability, 97
4.1.1 Series Systems, 97
4.1.2 Parallel Systems, 99
4.1.3 Standby Parallel, 102
4.2 Weibull Mixtures, 103
4.3 P(Y<X), 105
4.4 Radial Error, 108
contents ix
4.5 Pro Rata Warranty, 110
4.6 Optimum Age Replacement, 112
4.6.1 Age Replacement, 115
4.6.2 MTTF for a Maintained System, 117
4.7 Renewal Theory, 119
4.7.1 Block Replacement, 121
4.7.2 Free Replacement Warranty, 122
4.7.3 A Renewing Free Replacement Warranty, 122
4.8 Optimum Bidding, 123
4.9 Optimum Burn-In, 124
4.10 Spare Parts Provisioning, 126
References, 127
Exercises, 128
5. Estimation in Single Samples 130
5.1 Point and Interval Estimation, 130
5.2 Censoring, 130
5.3 Estimation Methods, 132
5.3.1 Menon’s Method, 132
5.3.2 An Order Statistic Estimate of x 134
0.10,
5.4 Graphical Estimation of Weibull Parameters, 136
5.4.1 Complete Samples, 136
5.4.2 Graphical Estimation in Censored Samples, 140
5.5 Maximum Likelihood Estimation, 145
5.5.1 The Exponential Distribution, 147
5.5.2 Confi dence Intervals for the Exponential Distribution—
Type II Censoring, 147
5.5.3 Estimation for the Exponential Distribution—Interval
Censoring, 150
5.5.4 Estimation for the Exponential Distribution—Type I
Censoring, 151
5.5.5 Estimation for the Exponential Distribution—The Zero
Failures Case, 153
5.6 ML Estimation for the Weibull Distribution, 154
5.6.1 Shape Parameter Known, 154
5.6.2 Confi dence Interval for the Weibull Scale Parameter—
Shape Parameter Known, Type II Censoring, 155
5.6.3 ML Estimation for the Weibull Distribution—Shape
Parameter Unknown, 157
x contents
5.6.4 Confi dence Intervals for Weibull Parameters—Complete
and Type II Censored Samples, 162
5.6.5 Interval Censoring with the Weibull, 167
5.6.6 Confi dence Limits for Weibull Parameters—Type I
Censoring, 167
References, 177
Exercises, 179
6. Sample Size Selection, Hypothesis Testing, and Goodness of Fit 180
6.1 Precision Measure for Maximum Likelihood (ML)
Estimates, 180
6.2 Interval Estimates from Menon’s Method of Estimation, 182
6.3 Hypothesis Testing—Single Samples, 184
6.4 Operating Characteristic (OC) Curves for One-Sided Tests of the
Weibull Shape Parameter, 188
6.5 OC Curves for One-Sided Tests on a Weibull Percentile, 191
6.6 Goodness of Fit, 195
6.6.1 Completely Specifi ed Distribution, 195
6.6.2 Distribution Parameters Not Specifi ed, 198
6.6.3 Censored Samples, 201
6.6.4 The Program ADStat, 201
6.7 Lognormal versus Weibull, 204
References, 210
Exercises, 212
7. The Program Pivotal.exe 213
7.1 Relationship among Quantiles, 216
7.2 Series Systems, 217
7.3 Confi dence Limits on Reliability, 218
7.4 Using Pivotal.exe for OC Curve Calculations, 221
7.5 Prediction Intervals, 224
7.6 Sudden Death Tests, 226
7.7 Design of Optimal Sudden Death Tests, 230
References, 233
Exercises, 234
8. Inference from Multiple Samples 235
8.1 Multiple Weibull Samples, 235
8.2 Testing the Homogeneity of Shape Parameters, 236
contents xi
8.3 Estimating the Common Shape Parameter, 238
8.3.1 Interval Estimation of the Common Shape
Parameter, 239
8.4 Interval Estimation of a Percentile, 244
8.5 Testing Whether the Scale Parameters Are Equal, 249
8.5.1 The SPR Test, 250
8.5.2 Likelihood Ratio Test, 252
8.6 Multiple Comparison Tests for Differences in Scale
Parameters, 257
8.7 An Alternative Multiple Comparison Test for Percentiles, 259
8.8 The Program Multi-Weibull.exe, 261
8.9 Inference on P (Y<X), 266
8.9.1 ML Estimation, 267
8.9.2 Normal Approximation, 269
8.9.3 An Exact Simulation Solution, 271
8.9.4 Confi dence Intervals, 273
References, 274
Exercises, 274
9. Weibull Regression 276
9.1 The Power Law Model, 276
9.2 ML Estimation, 278
9.3 Example, 279
9.4 Pivotal Functions, 280
9.5 Confi dence Intervals, 281
9.6 Testing the Power Law Model, 281
9.7 Monte Carlo Results, 282
9.8 Example Concluded, 285
9.9 Approximating u* at Other Stress Levels, 287
9.10 Precision, 289
9.11 Stress Levels in Different Proportions Than Tabulated, 289
9.12 Discussion, 291
9.13 The Disk Operating System (DOS) Program REGEST, 291
References, 296
Exercises, 296
10. The Three-Parameter Weibull Distribution 298
10.1 The Model, 298
10.2 Estimation and Inference for the Weibull Location
Parameter, 300