Stochastic Ageing and Dependence for Reliability

By: Chin-Diew Lai, Min Xie, Richard E. Barlow (Foreword by)

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Hardcover | 15 May 2006

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Ageing and dependence are two important characteristics in reliability and survival analysis, and they affect significantly the decision people make with regard to maintenance, repair/replacement, price setting, warranties, medical studies, and other areas. There are many papers published at different technical levels. This book aims at providing a state-of-the-art review of the subject so the interested readers may have a panoramic view of the theory and applications of the two areas.

This book serves as reference book for professors and researchers involved in reliability and survival analysis. Students with basic probability and statistics knowledge interested in applications will also find the book useful.

From the reviews:

"This book is an important addition to several well-written books on reliability theory and concepts. The book fulfills the authors aim in providing a comprehensive treatment of both ageing and dependence concepts, with emphasis on reliability and survival analysis....Overall, I recommend this book to all serious-minded researchers, practitioners, teachers, and graduate students who would like to have the most up-to-date information available on the subject of stochastic ageing and dependencies in reliability. It also serves as reference book on reliability and survival analysis and is a good addition to one's statistical or reliability library." Suprasad V. Amari, International Journal of Performability Engineering, Vol. 3, No. 2, April 2007

Industry Reviews

From the reviews:

"The book fulfills the authors aim in providing a comprehensive treatment of both ageing and dependence concepts ... . it contains a comprehensive list of references and provides up-to-date information on what has been written on the subject to date. Many important results of the last three decades are brought together and succintly summarized. ... I recommend this book to all serious minded researchers, practitioners, teachers, and graduate students who would like to have the most up-to-date information ... . " (Suprasad V. Amari, www.ijpe-online.com, October, 2006)

"This book provides a review of how systems age, with emphasis on the dependence properties between components of a system. ... The book will provide a useful reference work and would be good supplementary reading for a graduate course on reliability analysis." (D. J. Hand, Short Book Reviews, Vol. 26 (2), 2006)

"As the authors summarize, 'the book's emphasis is on reliability and survival analysis providing a comprehensive treatment for both ageing and dependence concepts'. ... This modern book, which has a firm basis in the theory of stochastic processes and statistical modeling, should be usefulfor reliability researchers as well as for practitioners, including students in reliability and applied probability." (Adriana Hornikova, Technometrics, Vol. 49 (2), 2007)

"This is a good book on mathematical theory for reliability. ... the book has a very broad introduction to concepts and models for rehability. ... This is an excellent reference book for researchers in the field of reliability. It covers material that is relatively up to date on stochastic ageing for the analysis of reliability." (Kailash C. Kapur, Mathematical Reviews, Issue 2007 h)

"[T]he book will be useful for theoretical reliability researchers. The book will also be useful for graduate students in reliability and applied probability as it provides an extensive list of references." (Biometrics, 64, March 2008)

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You Can Find This Book In

Non-Fiction Mathematics Applied Mathematics Stochastics Probability & Statistics Economics Econometrics Economic Statistics Computing & I.T.Computer Science Engineering & Technology Technology in General

Engineering in General Business & Management Operational Research Computer Hardware Mechanical Engineering & Materials Production & Industrial Engineering Industrial Quality Control

Introduction	p. 1
Aim and Scope of the Book	p. 1
Brief Overview	p. 2
Acronyms and Nomenclatures	p. 4
Concepts and Applications of Stochastic Ageing	p. 7
Introduction	p. 7
Characterizations of Lifetime Distributions	p. 9
Shape of a Failure Rate Function	p. 11
Ageing Distributions	p. 15
Exponential	p. 16
Gamma	p. 16
Truncated Normal	p. 17
Weibull	p. 18
Lognormal	p. 19
Birnbaum-Saunders	p. 20
Inverse Gaussian	p. 21
Gompertz	p. 22
Makeham	p. 22
Linear Failure Rate	p. 23
Lomax Distribution	p. 23
Log-logistic	p. 24
Burr XII	p. 25
Exponential-geometric (EG) and Generalization	p. 26
Basic Concepts for Univariate Reliability Classes	p. 27
Some Acronyms and Notions of Aging	p. 27
Definitions of Reliability Classes	p. 28
Interrelationships	p. 29
Properties of the Basic Ageing Classes	p. 31
Properties of IFR and DFR	p. 32
Properties of IFRA	p. 33
NBU and NBUE	p. 34
DMRL and IMRL	p. 38
Summary of Preservation Properties of Classes of Distributions	p. 38
Moments Inequalities	p. 39
Scaled TTT Transform and Characterizations of Ageing Classes	p. 42
Non-monotonic Failure Rates and Non-monotonic Mean Residual Lives	p. 44
Non-monotonic Failure Rates	p. 44
Non-monotonic Mean Residual Lives	p. 45
Some Further Classes of Ageing	p. 45
Failure Rates of Mixtures of Distributions	p. 47
Mixture of Two DFR Distributions	p. 48
Possible Shapes of r(t) When Two Subpopulations Are IFR	p. 48
Mixture of Two Gamma Densities with a Common Scale Parameter	p. 49
Mixture of Two Weibull Distributions	p. 50
Mixtures of Two Positively Truncated Normal Distributions	p. 52
Mixtures of Two Increasing Linear Failure Rate Distributions	p. 53
Mixtures of an IFR Distribution with an Exponential Distribution	p. 55
Failure Rate of Finite Mixture of Several Components Belonging to the Same Family	p. 56
Initial and Final Behavior of Failure Rates of Mixtures	p. 57
Continuous Mixtures of Distributions	p. 59
Partial Orderings and Generalized Partial Orderings	p. 60
Generlized Partial Orderings	p. 61
Connections Among the Partial Orderings	p. 64
Generalized Ageing Properties Classification	p. 64
Applications of Partial Orderings	p. 66
Relative Ageing	p. 67
Shapes of [eta] Function for s-order Equilibrium Distributions	p. 68
Concluding Remarks on Ageing	p. 70
Bathtub Shaped Failure Rate Life Distributions	p. 71
Introduction	p. 71
Bathtub Shaped Failure Rate Is Not a Myth	p. 72
Definitions and Basic Properties	p. 72
Acronyms for Bathtub Shaped Failure Rate Life Distributions	p. 73
Definitions	p. 74
Some Further Properties	p. 76
Families of Bathtub Shapes Failure Rate Distributions	p. 77
Bathtub Distributions with Explicit Failure Rate Functions	p. 77
Finite Range Distribution Families	p. 82
Bathtub Distributions with More Complicated Failure Rates	p. 84
A Mistaken Identity: the Mixed Weibull Family	p. 86
Some Comments on the Bathtub Shapes	p. 87
Construction Techniques for BT Distributions	p. 87
Glaser's Technique	p. 88
Convex Function	p. 88
Function of Random Variables	p. 88
Reliability and Stochastic Mechanisms	p. 88
Mixtures	p. 89
Sectional Models	p. 89
Polynomial of Finite Order	p. 90
TTT Transform	p. 90
Truncation of DFR Distribution	p. 90
Change Point Estimation for BT Distributions	p. 91
Mean Residual Life and Bathtub Shaped Life Distributions	p. 92
Mean Residual Life	p. 92
Bathtub Shaped Failure Rate and Decreasing Percentile Residual Life Function	p. 93
Relationships Among NWBUE, BT and IDMRL Classes	p. 93
Optimal Burn-in Time for Bathtub Distributions	p. 94
Concepts of Burn-in	p. 94
Burn-in and Bathtub Distributions	p. 95
Burn-in Time for BT Lifetime under Warranty Policies	p. 98
Optimal Replacement Time and Bathtub Shaped Failure Rate Distributions	p. 99
Upside-down Bathtub Shaped Failure Rate Distributions	p. 99
UBT Models	p. 100
Optimal Burn-in Decision for UBT Models	p. 102
Modified and Generalized Distributions	p. 102
Modified Bathtub Distributions	p. 102
Generalized Bathtub Curves	p. 104
Roller-Coaster Curves	p. 105
Applications	p. 106
Mean Residual Life - Concepts and Applications in Reliability Analysis	p. 109
Introduction	p. 109
Mean Residual Life and Other Ageing Properties	p. 110
Mean Residual Life and its Reciprocity with Failure Rate	p. 111
Mean Residual Lives of Some Well-known Lifetime Distributions	p. 112
Mean Residual Life Classes	p. 114
Monotonic MRL Classes	p. 114
Non-monotonic MRL Classes	p. 115
Non-monotonic MRL and Non-monotonic Failure Rate	p. 116
Non-monotonic Failure Rates Life Distribution	p. 117
Relations Between MRL and Failure Rate in Terms of Shapes and Locations of Their Change Points	p. 117
A General Approach Determining Shapes of Failure Rates and MRL Functions	p. 124
Roller-Coaster Failure Rates and Mean Residual Lives	p. 126
Effect of Burn-In on Mean Residual Life	p. 128
Optimal Burn-in Criteria	p. 130
Optimal Burn-in for Upside-down Bathtub Distributions	p. 130
Tests and Estimation of Mean Residual Life	p. 130
Tests for Monotonic Mean Residual Life	p. 131
Tests of Trend Change in Mean Residual Life	p. 131
Estimation of Monotonic Mean Residual Life	p. 131
Estimation of Change Points	p. 132
Mean Residual life with Special Characteristics	p. 132
Linear Mean Residual Life Function	p. 132
Proportional MRL and its Generalization	p. 133
Other Residual Life Functions	p. 133
Residual Life Distribution Function	p. 133
Variance Residual Life Function	p. 134
Percentile Residual Life Function	p. 134
Mean Residual Life Orderings	p. 134
Multivariate Mean Residual Life	p. 135
Characterizations of Multivariate Survival Distributions Based on Mean Residual Lives	p. 136
Bivariate Decreasing MRL	p. 137
Applications and Conclusions	p. 137
Weibull Related Distributions	p. 139
Introduction	p. 139
Basic Weibull Distribution	p. 140
Two-parameter Weibull Distribution and Basic Properties	p. 140
Parameter Estimation Methods	p. 142
Relative Ageing of Two 2-Parameter Weibull Distributions	p. 144
Three-parameter Weibull distribution	p. 144
Models Derived from Transformations of Weibull Variable	p. 148
Reflected Weibull Distribution	p. 148
Log Weibull Distribution	p. 149
Inverse (or Reverse) Weibull Model	p. 149
Modifications or Generalizations of Weibull Distribution	p. 150
Extended Weibull Distribution	p. 151
Exponentiated Weibull Distribution	p. 152
Modified Weibull Distribution	p. 154
Modified Weibull Extension	p. 155
Generalized Weibull Family	p. 156
Generalized Weibull Distribution of Gurvich et al	p. 158
Models Involving Two or More Weibull Distributions	p. 158
n-fold Mixture Model	p. 158
n-fold Competing Risk Model	p. 159
n-fold Multiplicative Model	p. 160
n-fold Sectional Model	p. 161
Model Involving Two Inverse Weibull Distributions	p. 161
Weibull Models with Varying Parameters	p. 162
Discrete Weibull Models	p. 163
Bivariate models	p. 163
Marshall and Olkin (1967)	p. 164
Lee (1979)	p. 164
Lu and Bhattacharyya (1990)-I	p. 164
Morgenstern-Gumbel-Farlie System	p. 165
Lu and Bhattacharyya (1990)-II	p. 165
Lee (1979)-II	p. 165
Applications of Weibull and Related Models	p. 165
An Introduction to Discrete Failure Time Models	p. 167
Introduction	p. 167
Survival Function, Failure Rate and Other Reliability Characteristics	p. 168
Elementary Ageing Classes	p. 171
IFR and DFR	p. 171
IFRA and DFRA	p. 174
NBU (NWU)	p. 175
NBUE	p. 175
DMRL and IMRL	p. 176
Relationships Among Discrete Ageing Concepts	p. 178
More Advanced Ageing Classes	p. 178
Non-monotonic Models	p. 179
BT Failure Rate and DIMRL	p. 180
UBT Failure Rate and DIMRL	p. 183
Discrete IDMRL (DIMRL) and BT (UBT) Failure Rate	p. 184
Discrete Bathtub-shaped Failure Rate Average	p. 186
Preservation under Poisson Shocks	p. 187
Examples of Discrete Time Failure Models	p. 187
Common Discrete Lifetime Distributions Derived from Continuous Ones	p. 188
Distributions Derived from Simple Failure Rate Functions	p. 191
Determination of Ageing from Ratio of Two Consecutive Probabilities	p. 192
Polya Urn Distributions	p. 194
Discussion on Discrete Failure Time Models	p. 195
Applications of Discrete Failure Time Models	p. 196
Some Problems of Usual Definition of Discrete Failure Rate	p. 198
Alternative Definition of Failure Rate and Its Ramification	p. 199
The Relationships between r(k) and r*(k)	p. 200
Effect of Alternative Failure Rate on Ageing Concepts	p. 200
Additive Property for Series System	p. 201
Examples	p. 202
Tests of Stochastic Ageing	p. 203
Introduction	p. 203
Exponential Distribution	p. 204
A General Sketch of Tests	p. 204
Estimation of Survival, Failure Rate and Mean Residual Life Functions	p. 206
Statistical Tests for Univariate Ageing Classes	p. 206
Some Common Bases for Test Statistics	p. 207
IFR Tests	p. 207
IFRA Tests	p. 209
NBU Tests	p. 211
NBUE Tests	p. 214
HNBUE	p. 215
NBU-t[subscript 0]	p. 216
NBUC Tests	p. 218
NBUFR (NWUFR) Test	p. 218
DPRL-[alpha] and NBUP-[alpha] Tests	p. 218
Summary of Tests of Basic Ageing Classes	p. 219
Tests of Aging Properties When Data Are Censored	p. 222
Tests of Monotonic Mean Residual Life Classes	p. 223
DMRL	p. 223
DMRLHA Test	p. 226
Tests of Non-monotonic Mean Residual Life	p. 226
IDMRL (DIMRL) Test When Turning Point [tau] Is Known	p. 228
IDMRL Test When the Proportion p Is Known	p. 228
Tests of IDMRL When Both p and [tau] Are Unkown	p. 229
Tests for NWBUE Class	p. 231
Tests of Exponentiality Versus Bathtub Distributions	p. 231
Test Based on Total Time on Test (TTT) Transform	p. 231
Park's Test for BT	p. 234
Graphical Tests for BT Failure Rate Distributions	p. 234
Other Miscellaneous Tests	p. 235
Test of Change Point of Failure Rate	p. 235
Aly's Tests for Change Point	p. 235
Testing Whether Lifetime Distribution Is Decreasing Uncertainty	p. 235
Final Remarks	p. 236
Bivariate and Multivariate Ageing	p. 237
Introduction	p. 237
Bivariate Reliability Classes	p. 238
Different Alternative Requirements	p. 238
Bivariate IFR	p. 239
Bivariate IFRA	p. 241
Bivariate NBU	p. 243
Bivariate NBUE and HNBUE	p. 244
Bivariate Decreasing Mean Residual Life	p. 245
Tests of Bivariate Ageing	p. 246
Summary on Tests of Bivariate Ageing	p. 249
Discrete Bivariate Failure Rates	p. 250
Applications	p. 251
Maintenance and Repairs	p. 251
Warranty Polices	p. 252
Failure Times of Pumps	p. 252
Bayesian Notions of Multivariate Ageing	p. 252
Motivations and Historical Development of Bayesian Approach	p. 253
Concepts of Ageing and Schur Concavity	p. 253
Bayesian Notions of Bivariate IFR	p. 254
Bayesian Bivariate DMRL	p. 256
Other Bayesian Bivariate Ageing Concepts	p. 257
Conclusions	p. 258
Concepts and Measures of Dependence in Reliability	p. 259
Introduction	p. 259
Important Conditions Describing Positive Dependence	p. 260
Six Basic Conditions	p. 261
The Relative Stringency of the Conditions	p. 263
Associated Random Variables	p. 264
RCSI and LCSD	p. 265
WPQD	p. 266
Positively Correlated Distributions	p. 266
Summary of Interrelationships	p. 266
Positive Quadrant Dependent (PQD) Concept	p. 267
Constructions of PQD Bivariate Distributions	p. 269
Applications of Positive Quadrant Dependence Concept to Reliability	p. 269
Families of Bivariate Distributions That Are PQD	p. 270
PQD Bivariate Distributions with Simple Structures	p. 271
PQD Bivariate Distributions with More Complicated Structures	p. 273
PQD Bivariate Uniform Distributions	p. 275
Some Related Issues on Bivariate Dependence	p. 277
Examples of Bivariate Positive Dependence Stronger than PQD Condition	p. 277
Examples of NQD and Other Negative Ageing	p. 280
Concluding Remarks on Concepts of Dependence	p. 281
Links Between Dependence Concepts and Bivariate Ageing Notions	p. 282
Dependence Concepts and Bayesian Multivariate Ageing	p. 283
Positive Dependence Orderings	p. 285
More PQD	p. 285
More SI	p. 286
More Associated	p. 287
More TP[subscript 2]	p. 288
Relations Among Different Partial Orderings	p. 288
Other Positive Dependence Orderings	p. 289
Multivariate Dependence Ordering	p. 289
Measures of Dependence	p. 290
Pearson's Product-Moment Correlation Coefficient	p. 291
Robustness of Sample Correlation	p. 292
Interpretation of Correlation	p. 293
Rank Correlations	p. 294
Kendall' tau	p. 295
Spearman's rho	p. 296
The Relationship between Kendall's tau and Spearman's rho	p. 298
Other Concordance Measures	p. 300
Local Measures of Dependence	p. 301
Definition of Local Dependence	p. 302
Local Dependence Function of Holland and Wang	p. 302
Properties of [gamma](x, y)	p. 303
Clayton-Oakes Association Measure	p. 304
Local [rho subscript S] and [tau]	p. 304
Local Correlation Coefficient	p. 305
Local Linear Dependence Function	p. 305
Applications of Several Local Indices in Survival Analysis	p. 306
Conclusion	p. 306
Reliability of Systems with Dependent Components	p. 307
Introduction	p. 307
Bivariate Distributions for Modelling Lifetimes of Two Components	p. 307
Examples of Bivariate Distributions Useful for Reliability Modelling	p. 309
Other Bivariate Distributions	p. 314
Effectiveness of Redundancy for Reliability System	p. 314
Redundancy	p. 314
Effectiveness of Parallel Redundancy of Two Independent and Identical Components	p. 315
Parallel Redundancy of Two Independent but Nonidentical Components	p. 316
Dependence Concepts and Redundancy	p. 316
Parallel Systems	p. 317
Mean Time to Failure of a Parallel System of Two Independent Components	p. 317
Mean Lifetime of a Parallel System with Two PQD Components	p. 317
Mean Lifetime of a Parallel System with Two NQD Components	p. 319
Relative Efficiency from Different Joint Distributions	p. 320
MTTF Comparisons of Three PQD Bivariate Exponential Distributions	p. 323
Efficiency of Redundancy by NQD Components	p. 324
Series Structures	p. 325
Series and Parallel System of n Positive Dependent Components	p. 326
Ageing Classes for Series and Parallel Systems with Two Dependent Components	p. 327
Ageing Class	p. 327
k-out-of-n Systems	p. 329
Reliability of a k-out-of-n System	p. 329
Ageing properties of a k-out-of-n system	p. 330
Comparative Studies of Two k-out-of-n Systems	p. 331
Ageing Properties Based on the Residual Life of a k-out-of-n System	p. 335
Dependent Component Lifetimes	p. 335
Consecutive k-out-of n:F Systems	p. 336
Reliability and Lifetime Distribution	p. 337
Structure Importance of Consecutive k-out-of-n Systems	p. 339
Algorithms for Determining Optimal Replacement Policies for Consecutive k-out-of-n Systems	p. 339
Ageing Property	p. 339
Consecutive-k-out-of-n:F System with Markov Dependence	p. 340
On Allocation of Spares to k-out-of-n Systems	p. 340
Standby Redundant System	p. 342
Standby Redundancy in k-out-of-n Systems	p. 342
Standby Redundancy at Component Versus System Level	p. 343
Dependent Components	p. 343
Future Directions	p. 344
Failure Time Data	p. 345
Introduction	p. 345
Empirical Modelling of Data	p. 345
Data Presentation and General Comments on Reliability Estimation	p. 346
IFR Data	p. 347
DFR Data	p. 349
NBU Data	p. 352
Bathtub Shaped Failure Rates Data	p. 353
Upside-down Bathtub Shaped Failure Rates Data	p. 358
Other Sources of Survival and Reliability Data	p. 362
References	p. 363
Index	p. 409
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Stochastic Ageing and Dependence for Reliability

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