Research Catalog

Title

Stochastic ageing and dependence for reliability / Chin-Diew Lai, Min Xie.

Author

Lai, C. D.

Publication

New York, NY : Springer Science + Business Media, [2006], ©2006.

Items in the Library & Off-site

Holdings

Details

Additional Authors

Xie, Min.

Description

xx, 418 pages : illustrations; 24 cm

Alternative Title

Stochastic aging and dependence for reliability

Subjects

• Reliability (Engineering) > Mathematical models
• Dependence (Statistics)
• Stochastic analysis
• Product life cycle > Mathematical models
• System failures (Engineering) > Mathematical models

Bibliography (note)

• Includes bibliographical references (p. [363]-408) and index.

Contents

• 1. Introduction -- 1.1. Aim and Scope of the Book -- 1.2. Brief Overview -- 1.3. Acronyms and Nomenclatures -- 2. Concepts and Applications of Stochastic Ageing -- 2.1. Introduction -- 2.2. Characterizations of Lifetime Distributions -- 2.2.1. Shape of a Failure Rate Function -- 2.3. Ageing Distributions -- 2.3.1. Exponential -- 2.3.2. Gamma -- 2.3.3. Truncated Normal -- 2.3.4. Weibull -- 2.3.5. Lognormal -- 2.3.6. Birnbaum-Saunders -- 2.3.7. Inverse Gaussian -- 2.3.8. Gompertz -- 2.3.9. Makeham -- 2.3.10. Linear Failure Rate -- 2.3.11. Lomax Distribution -- 2.3.12. Log-logistic -- 2.3.13. Burr XII -- 2.3.14. Exponential-geometric (EG) and Generalization -- 2.4. Basic Concepts for Univariate Reliability Classes -- 2.4.1. Some Acronyms and Notions of Aging -- 2.4.2. Definitions of Reliability Classes -- 2.4.3. Interrelationships -- 2.5. Properties of the Basic Ageing Classes -- 2.5.1. Properties of IFR and DFR -- 2.5.2. Properties of IFRA -- 2.5.3. NBU and NBUE -- 2.5.4. DMRL and IMRL -- 2.5.5. Summary of Preservation Properties of Classes of Distributions -- 2.5.6. Moments Inequalities -- 2.5.7. Scaled TTT Transform and Characterizations of Ageing Classes -- 2.6. Non-monotonic Failure Rates and Non-monotonic Mean Residual Lives -- 2.6.1. Non-monotonic Failure Rates -- 2.6.2. Non-monotonic Mean Residual Lives -- 2.7. Some Further Classes of Ageing -- 2.8. Failure Rates of Mixtures of Distributions -- 2.8.1. Mixture of Two DFR Distributions -- 2.8.2. Possible Shapes of r(t) When Two Subpopulations Are IFR -- 2.8.3. Mixture of Two Gamma Densities with a Common Scale Parameter -- 2.8.4. Mixture of Two Weibull Distributions -- 2.8.5. Mixtures of Two Positively Truncated Normal Distributions -- 2.8.6. Mixtures of Two Increasing Linear Failure Rate Distributions -- 2.8.7. Mixtures of an IFR Distribution with an Exponential Distribution -- 2.8.8. Failure Rate of Finite Mixture of Several Components Belonging to the Same Family -- 2.8.9. Initial and Final Behavior of Failure Rates of Mixtures -- 2.8.10. Continuous Mixtures of Distributions -- 2.9. Partial Orderings and Generalized Partial Orderings -- 2.9.1. Generlized Partial Orderings -- 2.9.2. Connections Among the Partial Orderings -- 2.9.3. Generalized Ageing Properties Classification -- 2.9.4. Applications of Partial Orderings -- 2.10. Relative Ageing -- 2.11. Shapes of [eta] Function for s-order Equilibrium Distributions -- 2.12. Concluding Remarks on Ageing -- 3. Bathtub Shaped Failure Rate Life Distributions -- 3.1. Introduction -- 3.2. Bathtub Shaped Failure Rate Is Not a Myth -- 3.3. Definitions and Basic Properties -- 3.3.1. Acronyms for Bathtub Shaped Failure Rate Life Distributions -- 3.3.2. Definitions -- 3.3.3. Some Further Properties -- 3.4. Families of Bathtub Shapes Failure Rate Distributions -- 3.4.1. Bathtub Distributions with Explicit Failure Rate Functions -- 3.4.2. Finite Range Distribution Families -- 3.4.3. Bathtub Distributions with More Complicated Failure Rates -- 3.4.4. A Mistaken Identity: the Mixed Weibull Family -- 3.4.5. Some Comments on the Bathtub Shapes -- 3.5. Construction Techniques for BT Distributions -- 3.5.1. Glaser's Technique -- 3.5.2. Convex Function -- 3.5.3. Function of Random Variables -- 3.5.4. Reliability and Stochastic Mechanisms -- 3.5.5. Mixtures -- 3.5.6. Sectional Models -- 3.5.7. Polynomial of Finite Order -- 3.5.8. TTT Transform -- 3.5.9. Truncation of DFR Distribution -- 3.6. Change Point Estimation for BT Distributions -- 3.7. Mean Residual Life and Bathtub Shaped Life Distributions -- 3.7.1. Mean Residual Life -- 3.7.2. Bathtub Shaped Failure Rate and Decreasing Percentile Residual Life Function -- 3.7.3. Relationships Among NWBUE, BT and IDMRL Classes -- 3.8. Optimal Burn-in Time for Bathtub Distributions -- 3.8.1. Concepts of Burn-in -- 3.8.2. Burn-in and Bathtub Distributions -- 3.8.3. Burn-in Time for BT Lifetime under Warranty Policies -- 3.8.4. Optimal Replacement Time and Bathtub Shaped Failure Rate Distributions -- 3.9. Upside-down Bathtub Shaped Failure Rate Distributions -- 3.9.1. UBT Models -- 3.9.2. Optimal Burn-in Decision for UBT Models -- 3.10. Modified and Generalized Distributions -- 3.10.1. Modified Bathtub Distributions -- 3.10.2. Generalized Bathtub Curves -- 3.10.3. Roller-Coaster Curves -- 3.11. Applications -- 4. Mean Residual Life - Concepts and Applications in Reliability Analysis -- 4.1. Introduction -- 4.2. Mean Residual Life and Other Ageing Properties -- 4.2.1. Mean Residual Life and its Reciprocity with Failure Rate -- 4.3. Mean Residual Lives of Some Well-known Lifetime Distributions -- 4.4. Mean Residual Life Classes -- 4.4.1. Monotonic MRL Classes -- 4.4.2. Non-monotonic MRL Classes -- 4.5. Non-monotonic MRL and Non-monotonic Failure Rate -- 4.5.1. Non-monotonic Failure Rates Life Distribution -- 4.5.2. Relations Between MRL and Failure Rate in Terms of Shapes and Locations of Their Change Points -- 4.5.3. A General Approach Determining Shapes of Failure Rates and MRL Functions -- 4.5.4. Roller-Coaster Failure Rates and Mean Residual Lives -- 4.6. Effect of Burn-In on Mean Residual Life -- 4.6.1. Optimal Burn-in Criteria -- 4.6.2. Optimal Burn-in for Upside-down Bathtub Distributions -- 4.7. Tests and Estimation of Mean Residual Life -- 4.7.1. Tests for Monotonic Mean Residual Life -- 4.7.2. Tests of Trend Change in Mean Residual Life -- 4.7.3. Estimation of Monotonic Mean Residual Life -- 4.7.4. Estimation of Change Points -- 4.8. Mean Residual life with Special Characteristics -- 4.8.1. Linear Mean Residual Life Function -- 4.8.2. Proportional MRL and its Generalization -- 4.9. Other Residual Life Functions -- 4.9.1. Residual Life Distribution Function -- 4.9.2. Variance Residual Life Function -- 4.9.3. Percentile Residual Life Function -- 4.10. Mean Residual Life Orderings -- 4.11. Multivariate Mean Residual Life -- 4.11.1. Characterizations of Multivariate Survival Distributions Based on Mean Residual Lives -- 4.11.2. Bivariate Decreasing MRL -- 4.12. Applications and Conclusions -- 5. Weibull Related Distributions -- 5.1. Introduction -- 5.2. Basic Weibull Distribution -- 5.2.1. Two-parameter Weibull Distribution and Basic Properties -- 5.2.2. Parameter Estimation Methods -- 5.2.3. Relative Ageing of Two 2-Parameter Weibull Distributions -- 5.3. Three-parameter Weibull distribution -- 5.4. Models Derived from Transformations of Weibull Variable -- 5.4.1. Reflected Weibull Distribution -- 5.4.2. Log Weibull Distribution -- 5.4.3. Inverse (or Reverse) Weibull Model -- 5.5. Modifications or Generalizations of Weibull Distribution -- 5.5.1. Extended Weibull Distribution -- 5.5.2. Exponentiated Weibull Distribution -- 5.5.3. Modified Weibull Distribution -- 5.5.4. Modified Weibull Extension -- 5.5.5. Generalized Weibull Family -- 5.5.6. Generalized Weibull Distribution of Gurvich et al -- 5.6. Models Involving Two or More Weibull Distributions -- 5.6.1. n-fold Mixture Model -- 5.6.2. n-fold Competing Risk Model -- 5.6.3. n-fold Multiplicative Model -- 5.6.4. n-fold Sectional Model -- 5.6.5. Model Involving Two Inverse Weibull Distributions -- 5.7. Weibull Models with Varying Parameters -- 5.8. Discrete Weibull Models -- 5.9. Bivariate models -- 5.9.1. Marshall and Olkin (1967) -- 5.9.2. Lee (1979) -- 5.9.3. Lu and Bhattacharyya (1990)-I -- 5.9.4. Morgenstern-Gumbel-Farlie System -- 5.9.5. Lu and Bhattacharyya (1990)-II -- 5.9.6. Lee (1979)-II -- 5.10. Applications of Weibull and Related Models -- 6. An Introduction to Discrete Failure Time Models -- 6.1. Introduction -- 6.2. Survival Function, Failure Rate and Other Reliability Characteristics -- 6.3. Elementary Ageing Classes --
• 6.3.1. IFR and DFR -- 6.3.2. IFRA and DFRA -- 6.3.3. NBU (NWU) -- 6.3.4. NBUE -- 6.3.5. DMRL and IMRL -- 6.3.6. Relationships Among Discrete Ageing Concepts -- 6.4. More Advanced Ageing Classes -- 6.5. Non-monotonic Models -- 6.5.1. BT Failure Rate and DIMRL -- 6.5.2. UBT Failure Rate and DIMRL -- 6.5.3. Discrete IDMRL (DIMRL) and BT (UBT) Failure Rate -- 6.5.4. Discrete Bathtub-shaped Failure Rate Average -- 6.6. Preservation under Poisson Shocks -- 6.7. Examples of Discrete Time Failure Models -- 6.7.1. Common Discrete Lifetime Distributions Derived from Continuous Ones -- 6.7.2. Distributions Derived from Simple Failure Rate Functions -- 6.7.3. Determination of Ageing from Ratio of Two Consecutive Probabilities -- 6.7.4. Polya Urn Distributions -- 6.8. Discussion on Discrete Failure Time Models -- 6.9. Applications of Discrete Failure Time Models -- 6.10. Some Problems of Usual Definition of Discrete Failure Rate -- 6.11. Alternative Definition of Failure Rate and Its Ramification -- 6.11.1. The Relationships between r(k) and r*(k) -- 6.11.2. Effect of Alternative Failure Rate on Ageing Concepts -- 6.11.3. Additive Property for Series System -- 6.11.4. Examples -- 7. Tests of Stochastic Ageing -- 7.1. Introduction -- 7.2. Exponential Distribution -- 7.3. A General Sketch of Tests -- 7.3.1. Estimation of Survival, Failure Rate and Mean Residual Life Functions -- 7.4. Statistical Tests for Univariate Ageing Classes -- 7.4.1. Some Common Bases for Test Statistics -- 7.4.2. IFR Tests -- 7.4.3. IFRA Tests -- 7.4.4. NBU Tests -- 7.4.5. NBUE Tests -- 7.4.6. HNBUE -- 7.4.7. NBU-t[subscript 0] -- 7.4.8. NBUC Tests -- 7.4.9. NBUFR (NWUFR) Test -- 7.4.10. DPRL-[alpha] and NBUP-[alpha] Tests -- 7.4.11. Summary of Tests of Basic Ageing Classes -- 7.5. Tests of Aging Properties When Data Are Censored -- 7.6. Tests of Monotonic Mean Residual Life Classes -- 7.6.1. DMRL -- 7.6.2. DMRLHA Test -- 7.7. Tests of Non-monotonic Mean Residual Life -- 7.7.1. IDMRL (DIMRL) Test When Turning Point [tau] Is Known -- 7.7.2. IDMRL Test When the Proportion p Is Known -- 7.7.3. Tests of IDMRL When Both p and [tau] Are Unknown -- 7.7.4. Tests for NWBUE Class -- 7.8. Tests of Exponentiality Versus Bathtub Distributions -- 7.8.1. Test Based on Total Time on Test (TTT) Transform -- 7.8.2. Park's Test for BT -- 7.8.3. Graphical Tests for BT Failure Rate Distributions -- 7.9. Other Miscellaneous Tests -- 7.9.1. Test of Change Point of Failure Rate -- 7.9.2. Aly's Tests for Change Point -- 7.9.3. Testing Whether Lifetime Distribution Is Decreasing Uncertainty -- 7.10. Final Remarks -- 8. Bivariate and Multivariate Ageing -- 8.1. Introduction -- 8.2. Bivariate Reliability Classes -- 8.2.1. Different Alternative Requirements -- 8.3. Bivariate IFR -- 8.4. Bivariate IFRA -- 8.5. Bivariate NBU -- 8.6. Bivariate NBUE and HNBUE -- 8.7. Bivariate Decreasing Mean Residual Life -- 8.8. Tests of Bivariate Ageing -- 8.8.1. Summary on Tests of Bivariate Ageing -- 8.9. Discrete Bivariate Failure Rates -- 8.10. Applications -- 8.10.1. Maintenance and Repairs -- 8.10.2. Warranty Polices -- 8.10.3. Failure Times of Pumps -- 8.11. Bayesian Notions of Multivariate Ageing -- 8.11.1. Motivations and Historical Development of Bayesian Approach -- 8.11.2. Concepts of Ageing and Schur Concavity -- 8.11.3. Bayesian Notions of Bivariate IFR -- 8.11.4. Bayesian Bivariate DMRL -- 8.11.5. Other Bayesian Bivariate Ageing Concepts -- 8.12. Conclusions -- 9. Concepts and Measures of Dependence in Reliability -- 9.1. Introduction -- 9.2. Important Conditions Describing Positive Dependence -- 9.2.1. Six Basic Conditions -- 9.2.2. The Relative Stringency of the Conditions -- 9.2.3. Associated Random Variables -- 9.2.4. RCSI and LCSD -- 9.2.5. WPQD -- 9.2.6. Positively Correlated Distributions -- 9.2.7. Summary of Interrelationships -- 9.3. Positive Quadrant Dependent (PQD) Concept -- 9.3.1. Constructions of PQD Bivariate Distributions -- 9.3.2. Applications of Positive Quadrant Dependence Concept to Reliability -- 9.4. Families of Bivariate Distributions That Are PQD -- 9.4.1. PQD Bivariate Distributions with Simple Structures -- 9.4.2. PQD Bivariate Distributions with More Complicated Structures -- 9.4.3. PQD Bivariate Uniform Distributions -- 9.5. Some Related Issues on Bivariate Dependence -- 9.5.1. Examples of Bivariate Positive Dependence Stronger than PQD Condition -- 9.5.2. Examples of NQD and Other Negative Ageing -- 9.5.3. Concluding Remarks on Concepts of Dependence -- 9.6. Links Between Dependence Concepts and Bivariate Ageing Notions -- 9.7. Dependence Concepts and Bayesian Multivariate Ageing -- 9.8. Positive Dependence Orderings -- 9.8.1. More PQD -- 9.8.2. More SI -- 9.8.3. More Associated -- 9.8.4. More TP[subscript 2] -- 9.8.5. Relations Among Different Partial Orderings -- 9.8.6. Other Positive Dependence Orderings -- 9.8.7. Multivariate Dependence Ordering -- 9.9. Measures of Dependence -- 9.10. Pearson's Product-Moment Correlation Coefficient -- 9.10.1. Robustness of Sample Correlation -- 9.10.2. Interpretation of Correlation -- 9.11. Rank Correlations -- 9.11.1. Kendall' tau -- 9.11.2. Spearman's rho -- 9.11.3. The Relationship between Kendall's tau and Spearman's rho -- 9.11.4. Other Concordance Measures -- 9.12. Local Measures of Dependence -- 9.12.1. Definition of Local Dependence -- 9.12.2. Local Dependence Function of Holland and Wang -- 9.12.3. Properties of [gamma](x, y) -- 9.12.4. Clayton-Oakes Association Measure -- 9.12.5. Local [rho subscript S] and [tau] -- 9.12.6. Local Correlation Coefficient -- 9.12.7. Local Linear Dependence Function -- 9.12.8. Applications of Several Local Indices in Survival Analysis -- 9.13. Conclusion -- 10. Reliability of Systems with Dependent Components -- 10.1. Introduction -- 10.2. Bivariate Distributions for Modelling Lifetimes of Two Components -- 10.2.1. Examples of Bivariate Distributions Useful for Reliability Modelling -- 10.2.2. Other Bivariate Distributions -- 10.3. Effectiveness of Redundancy for Reliability System -- 10.3.1. Redundancy -- 10.3.2. Effectiveness of Parallel Redundancy of Two Independent and Identical Components -- 10.3.3. Parallel Redundancy of Two Independent but Nonidentical Components -- 10.3.4. Dependence Concepts and Redundancy -- 10.4. Parallel Systems -- 10.4.1. Mean Time to Failure of a Parallel System of Two Independent Components -- 10.4.2. Mean Lifetime of a Parallel System with Two PQD Components -- 10.4.3. Mean Lifetime of a Parallel System with Two NQD Components -- 10.4.4. Relative Efficiency from Different Joint Distributions -- 10.4.5. MTTF Comparisons of Three PQD Bivariate Exponential Distributions -- 10.4.6. Efficiency of Redundancy by NQD Components -- 10.5. Series Structures -- 10.5.1. Series and Parallel System of n Positive Dependent Components -- 10.6. Ageing Classes for Series and Parallel Systems with Two Dependent Components -- 10.6.1. Ageing Class -- 10.7. k-out-of-n Systems -- 10.7.1. Reliability of a k-out-of-n System -- 10.7.2. Ageing properties of a k-out-of-n system -- 10.7.3. Comparative Studies of Two k-out-of-n Systems -- 10.7.4. Ageing Properties Based on the Residual Life of a k-out-of-n System -- 10.7.5. Dependent Component Lifetimes -- 10.8. Consecutive k-out-of n:F Systems -- 10.8.1. Reliability and Lifetime Distribution -- 10.8.2. Structure Importance of Consecutive k-out-of-n Systems -- 10.8.3. Algorithms for Determining Optimal Replacement Policies for Consecutive k-out-of-n Systems -- 10.8.4. Ageing Property -- 10.8.5. Consecutive-k-out-of-n:F System with Markov Dependence -- 10.9. On Allocation of Spares to k-out-of-n Systems -- 10.10. Standby Redundant System --
• 10.10.1. Standby Redundancy in k-out-of-n Systems -- 10.10.2. Standby Redundancy at Component Versus System Level -- 10.10.3. Dependent Components -- 10.11. Future Directions -- 11. Failure Time Data -- 11.1. Introduction -- 11.2. Empirical Modelling of Data -- 11.3. Data Presentation and General Comments on Reliability Estimation -- 11.4. IFR Data -- 11.5. DFR Data -- 11.6. NBU Data -- 11.7. Bathtub Shaped Failure Rates Data -- 11.8. Upside-down Bathtub Shaped Failure Rates Data -- 11.9. Other Sources of Survival and Reliability Data.

ISBN

0387297421

LCCN

9780387297422

OCLC

• ocm69652023
• SCSB-5261649

Owning Institutions

Columbia University Libraries