Recent Advances in Reliability Theory: Methodology, Practice, and Inference

514 Pages·2000·44.865 MB·English

by  Igor Ushakov (auth.), N. Limnios, M. Nikulin (eds.)

Statistics for Industry and Technology
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Recent Advances in
Reliability Theory
Methodology, Practice,
and Inference
N. Limnios
M. Nikulin
Editors
Springer Science+Business Media, LLC
N. Limnios M. Nikulin
Division Mathematiques Appliquees Universite Victor Segalen-Bordeaux 2
Universite de Technologie de Compiegne Bordeaux Cedex 33076
Compiegne Cedex 60205 France
France
Library of Congress Cataloging-in-Publication Data
Recent advances in reliability theory : methodology, practice, and inference I editors,
N. Limnios, M. Nikulin.
p. em. - (Statistics for industry and technology)
Includes bibliographical references.
ISBN 978-1-4612-7124-6 ISBN 978-1-4612-1384-0 (eBook)
DOI 10.1007/978-1-4612-1384-0
1. Reliability (Engineering)-Mathematical models. 2. Stochastic processes.
1. Limnios, N. (Nikolaos) II. Nikulin, M.S. (Mikhail Stepanovich) III. Series.
TA169 .R42 2000
620'.00452-dc21 00-039790
CIP
Prlnted on acid-free paper.
© 2000 Springer Science+Business Media New York
Originally published by Birkhlluser Boston in 2000
Softcover reprint of the hardcover 1s t edition 2000
AlI rights reserved. This work may not be translated or copied in whole or in part without the written permission
of the publisher Springer Science+Business Media, LLC,
except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with
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dise Marks Act, may accordingly be used freely by anyone.
ISBN 978-1-4612-7124-6 SPIN 10727141
Typeset by the editors in LaTeX.
Cover design by Vemon Press, Boston, MA.
9 8 7 6 5 432 1
Contents
Preface xv
Contributors xvii
List of Tables xxiii
List ofFigures xxv
I General Approach 1
1 Reliability: Past, Present, Future 3
Igor Ushakov
1.1 Introduction.................... 3
1.2 Main Directions of Modern Reliability Theory . 4
1.3 History of Ideas in Reliability . . . . . . . . . 7
1.4 In the White Water ofa Publications Stream 11
1.5 Problems Expecting Solutions . 12
1.6 Conclusion .. 14
2 Reliability Analysis as a Tool for Expressing and
Communicating Uncertainty 23
Terje Aven
2.1 Introduction. 23
2.2 Basic Principles . 25
2.3 Examples . . . . 27
2.3.1 Production loss . 27
2.3.2 Methods ofstructural reliability analysis. 32
2.4 Discussion.......... . 33
v
vi Contents
II Probability Models and Related Issues 39
3 Modeling a Process of Non-Ideal Repair 41
Max S. Finkelstein
3.1 Introduction..................... 41
3.2 Imperfect and General Repair: A BriefReview. 43
3.3 Multiphase ALM. 46
3.4 Final Remarks . 50
4 Some Models and Mathematical Results for Reliability of
Systems of Components 55
Christiane Cocozza-Thivent
4.1 Which Model to Choose? . . . . . . . . . . . . 55
4.1.1 Several models . . . . . . . . . . . . . . 55
4.1.2 Comparisons with constant failure rates 57
4.2 Comparison of Systems 60
4.2.1 Comparison in case ofMarkov models 60
4.2.2 Comparison in case ofindependent components with
general rates .... . . . . . . . 61
4.3 Results Depending on the Mean Values 61
4.3.1 For a semi-Markov process 61
4.3.2 For a product of independent semi-Markov processes. 61
4.4 Some Explicit Formulas . . . . . . . . . . . . . . . . . . . . . 62
4.4.1 General results for a product of interacting
semi-Markov processes. . . . . . . . . . .. 62
4.4.2 Case of a product ofindependent alternating
renewal processes . 62
4.5 Technics ofProof . . . . . . . . . . . . . 64
4.5.1 Renewal theory. . . . . . . . . . 64
4.5.2 Adding supplementary variables 65
4.5.3 Martingale technics 66
5 Algorithms of Stochastic Activity and Problems ofReliability 69
Boris Harlamov
5.1 Introduction...................... 69
5.2 Determinate Dependence on a Valuating Function 71
5.3 Random Dependence on a Valuating Function. 74
5.4 Algorithm of Search of Minimum Intensity of a
Stream ofRefusals . . . . . . . . . . . . . . . . . . . . . .. .. 80
6 Some Shifted Stochastic Orders 85
Rosa E. Lillo, Asok K. Nanda and Moshe Shaked
Contents vii
6.1 Introduction and Preliminaries . . . . . . . . . . 85
6.1.1 Logconcavity and logconvexity revisited . 85
6.1.2 The likelihood ratio and the hazard rate
orders revisited . . . . . . . . 87
6.1.3 Some facts on order statistics 88
6.2 The Up Likelihood Ratio Order .. 89
6.3 The Down Likelihood Ratio Order 93
6.4 The Up Hazard Rate Order .. 97
6.5 The Down Hazard Rate Order .. · 100
7 Characterization of Distributions in Reliability 105
Lev Klebanov and Gabor Szekely
7.1 Introduction . · 105
7.2 The Main Equation and its Analytic Solutions · 107
7.3 The General Solution ofthe Main Equation .. · 109
7.4 Other Normalizations . · 111
7.5 Reconstruction of the Reliability Polynomial from the
Limit Distribution . . . . . . . . . . . . · 112
7.6 A Generalization ofthe Main Equation. · 112
7.7 A General Limit Result . · 114
III Asymptotic Analysis 117
8 Asymptotic Analysis of Reliability for Switching
Systems in Light and Heavy Traffic Conditions 119
Vladimir V. Anisimov
8.1 Introduction................... . 119
8.2 Flows ofRare Events in Systems with Mixing . 121
8.3 Asymptotically Connected Sets (S-Sets) . 123
8.3.1 Homogeneous case . . . . . . . . 123
8.3.2 Nonhomogeneous case . . . . . . 125
8.4 Asymptotically Consolidated Systems . 126
8.5 Heavy Traffic Conditions. . . . . . . . . 127
8.6 Analysis ofReliability of Queueing Models. . 129
8.6.1 Light traffic analysis in models with finite capacity . 129
8.6.2 Heavy traffic analysis . 130
8.6.3 Systems with highly reliable servers ..... . 131
9 Nonlinearly Perturbed Markov Chains and Large
Deviations for Lifetime Functionals 135
Dmitrii S. Silvestrov
9.1 Mixed Large Deviation and Quasi-Ergodic Theorems. · 135
Contents
Vlll
9.2 Asymptotical Expansions for mgfofFirst Hitting Times . .. . 138
9.3 Asymptotical Expansions for Quasi-Stationary Distributions. . 139
9.4 Pseudo- and Quasi-Stationary Asymptotics . . . . . . . . .. . 140
10 Evolutionary Systems in an Asymptotic Split Phase Space 145
Vladimir S. Korolyuk and Nikolaos Limnios
10.1 Introduction 145
10.2 Representation ofa Stochastic Evolutionary
System. . . . . . . . . . . . . . . . . . . . . . . 146
10.3 Examples of Stochastic Evolutionary Systems . 148
10.4 Phase Merging Scheme. . . . . . . 148
10.5 Average Scheme . 152
10.6 Diffusion Approximation Scheme . 155
10.7 Reliability Interpretation. . . . . . 158
11 An Asymptotic Approach to Multistate Systems
Reliability Evaluation 163
Krzysztof Kolowrocki
11.1 Introduction. . . . . . . . . . . . . . . . . . . . . 163
11.2 Limit Reliability Functions of Series Systems . . 168
11.3 Limit Reliability Functions ofParallel Systems . 172
11.4 Applications. . . . . . . . . . . . . . . . . . . . . 175
IV Statistical Models and Data Analysis 181
12 Computer Intensive Methods Based on Resampling
in Analysis ofReliability and Survival Data 183
Yuri K. Belyaev
12.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 183
12.2 Weakly Approaching Distributions . . . . . . . . . . . . 185
12.3 Applications to Assessing Distributions ofEstimators. . 188
12.4 Assessing Distributions of Maximum Likelihood Estimators . 194
12.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .. . 196
13 Statistical Analysis of Damage Processes 199
Waltraud Kahle and Heide Wendt
13.1 Introduction. . . . . . . . . . . . . . . . . . 199
13.2 Damage Models. . . . . . . . . . . . . . . . 201
13.2.1 Models based on continuous processes . 201
13.2.2 Shock models . . . . . . . . . . . . . . . 203
13.3 Modeling Damage by Marked Point Processes . . 203
Contents ix
14 Data Analysis Based on Warranty Database 213
Kazuyuki Suzuki, Wataru Yamamoto, Md. Rezaul Karim
and Lianhua Wang
14.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . 213
14.2 Age-based Claim Analysis - Marginal Count Data. . 215
14.2.1 Repairable products . . . . . . . . . . . . . . . 216
14.2.2 Nonrepairable products - Poisson approximation . 218
14.2.3 An example . . . . . . . . . . . . . . . . 219
14.3 Estimation ofFailure Time Distribution . . . . 220
14.4 Analysis Without Sales Amount Information . 224
14.5 Conclusion . 225
15 Failure Models Indexed by Time and Usage 229
Simon P. Wilson
15.1 Introduction and Overview . 229
15.2 Formulation ofFailure Models Indexed by Two Scales . 231
15.2.1 Modelling the time-dependent covariate . . . . . 231
15.2.2 Modelling the effect ofthe covariate on time to failure . 235
15.3 Examples . . . . . . . . . . . . . . . . . . . . . 238
15.3.1 The Poisson process . . . . . . . . . . . 239
15.3.2 The doubly stochastic Poisson process . 239
15.3.3 The compound Poisson process . . . . . 240
16 A New Multiple ProofLoads Approach For
Estimating Correlations 245
Richard Johnson and Wenqing Lu
16.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 245
16.2 Review of Current Designs for Estimating Correlation . 246
16.3 A New Double ProofLoad Design . 249
16.3.1 Design, notation and the likelihood. . . . . . . . 249
16.3.2 The Fisher information for the double proofloading . 250
16.3.3 Numerical results for the Fisher information. . . . 252
16.4 A Symmetric Version ofthe Double ProofLoad Design. . 253
16.5 Asymptotic Results. . . . . . . . . . . . . . . . . . . . . 254
16.6 An Example Using the Bivariate Weibull Distribution . 255
16.7 Conclusion . 256
17 Conditional and Partial Correlation For Graphical
Uncertainty Models 259
Dorota Kurowicka and Roger Cooke
17.1 Introduction. . . . . . . . . . . . 259
17.2 Partial and Conditional Correlation . 260
x Contents
17.3 Trees, Vines and Copulas .262
17.4 Conditions for ~ = 0 . .264
17.5 Copulae . .266
17.6 Computing ~ . .269
17.7 Numerical Results .273
17.8 Conclusions .... .275
V Methods Common to Reliability and Survival Analysis 277
18 Semiparametric Methods of Time Scale Selection 279
Thierry Duchesne
18.1 Introduction. . . . . . . . . 279
18.2 Modeling Failure . . . . . . 280
18.3 Semiparametric Inference . 282
18.4 Semiparametric Model Assessment . 285
18.5 Nonparametric Age Curves . 286
18.6 Conclusion . 288
19 Censored and Truncated Lifetime Data 291
Catherine Huber
19.1 Introduction. . . . . . . . . . . . . . . . . 291
19.2 Examples of Censoring and Truncation. . 293
19.2.1 Censoring . . . . . . . . . . . . . . 293
19.2.2 Truncation .;.......... . 293
19.3 Regression Models: Independent Case. . 294
19.3.1 Cox model for independent individuals. . 294
19.3.2 Logistic discrete regression models . 296
19.3.3 Accelerated models. . . . . 296
19.4 Models for Clustered Data. . . . . . . . . . 297
19.4.1 Dependence as a nuisance . . . . . . 298
19.4.2 Models for estimating dependence . 300
20 Tests for a Family of Survival Models Based on Extremes 307
Martin Crowder
20.1 The Situation. . . . . . . . . . . . . 307
20.2 Distributions of Sample Extremes. . 309
20.3 A Test for 1/ = 0 . 310
=
20.4 A Test for 1 . 312
1/
20.5 A Test for K, = 0 . 313
=
20.6 A Test for T 0 . 313
= =
20.7 Tests for 0 . 314
1/ T
= =
20.8 Tests for 1/ - 1 T 0 . . 315