Recent Advances in Reliability Theory : Methodology, Practice and Interference

by Limnios, N.; Nikulin, M. S.

• ISBN13: 9780817641351
• ISBN10: 0817641351
• Format: Hardcover
• Copyright: 2000-07-01
• Publisher: Springer Verlag

Summary

This book presents thirty-one extensive and carefully edited chapters providing an up-to-date survey of new models and methods for reliability analysis and applications in science, engineering, and technology. The chapters contain broad coverage of the latest developments and innovative techniques in a wide range of theoretical and numerical issues in the field of statistical and probabilistic methods in reliability.

Table of Contents

Preface	xv
Contributors	xvii
List of Tables	xxiii
List of Figures	xxv
I General Approach	1	(38)
Reliability: Past, Present, Future	3	(20)
Igor Ushakov
Introduction	3	(1)
Main Directions of Modern Reliability Theory	4	(3)
History of Ideas in Reliability	7	(4)
In the White Water of a Publications Stream	11	(1)
Problems Expecting Solutions	12	(2)
Conclusion	14	(9)
Reliability Analysis as a Tool for Expressing and Communicating Uncertainty	23	(16)
Terje Aven
Introduction	23	(2)
Basic Principles	25	(2)
Examples	27	(6)
Production loss	27	(5)
Methods of structural reliability analysis	32	(1)
Discussion	33	(6)
II Probability Models and Related Issues	39	(78)
Modeling a Process of Non-Ideal Repair	41	(14)
Max S. Finkelstein
Introduction	41	(2)
Imperfect and General Repair: A Brief Review	43	(3)
Multiphase ALM	46	(4)
Final Remarks	50	(5)
Some Models and Mathematical Results for Reliability of Systems of Components	55	(14)
Christiane Cocozza-Thivent
Which Model to Choose?	55	(5)
Several models	55	(2)
Comparisons with constant failure rates	57	(3)
Comparison of Systems	60	(1)
Comparison in case of Markov models	60	(1)
Comparison in case of independent components with general rates	61	(1)
Results Depending on the Mean Values	61	(1)
For a semi-Markov process	61	(1)
For a product of independent semi-Markov processes	61	(1)
Some Explicit Formulas	62	(2)
General results for a product of interacting semi-Markov processes	62	(1)
Case of a product of independent alternating renewal processes	62	(2)
Technics of Proof	64	(5)
Renewal theory	64	(1)
Adding supplementary variables	65	(1)
Martingale technics	66	(3)
Algorithms of Stochastic Activity and Problems of Reliability	69	(16)
Boris Harlamov
Introduction	69	(2)
Determinate Dependence on a Valuating Function	71	(3)
Random Dependence on a Valuating Function	74	(6)
Algorithm of Search of Minimum Intensity of a Stream of Refusals	80	(5)
Some Shifted Stochastic Orders	85	(20)
Rosa E. Lillo
Asok K. Nanda
Moshe Shaked
Introduction and Preliminaries	85	(4)
Logconcavity and logconvexity revisited	85	(2)
The likelihood ratio and the hazard rate orders revisited	87	(1)
Some facts on order statistics	88	(1)
The Up Likelihood Ratio Order	89	(4)
The Down Likelihood Ratio Order	93	(4)
The Up Hazard Rate Order	97	(3)
The Down Hazard Rate Order	100	(5)
Characterization of Distributions in Reliability	105	(12)
Lev Klebanov
Gabor Szekely
Introduction	105	(2)
The Main Equation and its Analytic Solutions	107	(2)
The General Solution of the Main Equation	109	(2)
Other Normalizations	111	(1)
Reconstruction of the Reliability Polynomial from the Limit Distribution	112	(1)
A Generalization of the Main Equation	112	(2)
A General Limit Result	114	(3)
III Asymptotic Analysis	117	(64)
Asymptotic Analysis of Reliability for Switching Systems in Light and Heavy Traffic Conditions	119	(16)
Vladimir V. Anisimov
Introduction	119	(2)
Flows of Rare Events in Systems with Mixing	121	(2)
Asymptotically Connected Sets (S-Sets)	123	(3)
Homogeneous case	123	(2)
Nonhomogeneous case	125	(1)
Asymptotically Consolidated Systems	126	(1)
Heavy Traffic Conditions	127	(2)
Analysis of Reliability of Queueing Models	129	(6)
Light traffic analysis in models with finite capacity	129	(1)
Heavy traffic analysis	130	(1)
Systems with highly reliable servers	131	(4)
Nonlinearly Perturbed Markov Chains and Large Deviations for Lifetime Functionals	135	(10)
Dmitrii S. Silvestrov
Mixed Large Deviation and Quasi-Ergodic Theorems	135	(3)
Asymptotical Expansions for mgf of First Hitting Times	138	(1)
Asymptotical Expansions for Quasi-Stationary Distributions	139	(1)
Pseudo- and Quasi-Stationary Asymptotics	140	(5)
Evolutionary Systems in an Asymptotic Split Phase Space	145	(18)
Vladimir S. Korolyuk
Nikolaos Limnios
Introduction	145	(1)
Representation of a Stochastic Evolutionary System	146	(2)
Examples of Stochastic Evolutionary Systems	148	(1)
Phase Merging Scheme	148	(4)
Average Scheme	152	(3)
Diffusion Approximation Scheme	155	(3)
Reliability Interpretation	158	(5)
An Asymptotic Approach to Multistate Systems Reliability Evaluation	163	(18)
Krzysztof Kolowrocki
Introduction	163	(5)
Limit Reliability Functions of Series Systems	168	(4)
Limit Reliability Functions of Parallel Systems	172	(3)
Applications	175	(6)
IV Statistical Models and Data Analysis	181	(96)
Computer Intensive Methods Based on Resampling in Analysis of Reliability and Survival Data	183	(16)
Yuri K. Belyaev
Introduction	183	(2)
Weakly Approaching Distributions	185	(3)
Applications to Assessing Distributions of Estimators	188	(6)
Assessing Distributions of Maximum Likelihood Estimators	194	(2)
Conclusions	196	(3)
Statistical Analysis of Damage Processes	199	(14)
Waltraud Kahle
Heide Wendt
Introduction	199	(2)
Damage Models	201	(2)
Models based on continuous processes	201	(2)
Shock models	203	(1)
Modeling Damage by Marked Point Processes	203	(10)
Data Analysis Based on Warranty Database	213	(16)
Kazuyuki Suzuki
Wataru Yamamoto
Md. Rezaul Karim
Lianhua Wang
Introduction	213	(2)
Age-based Claim Analysis --- Marginal Count Data	215	(5)
Repairable products	216	(2)
Nonrepairable products - Poisson approximation	218	(1)
An example	219	(1)
Estimation of Failure Time Distribution	220	(4)
Analysis Without Sales Amount Information	224	(1)
Conclusion	225	(4)
Failure Models Indexed by Time and Usage	229	(16)
Simon P. Wilson
Introduction and Overview	229	(2)
Formulation of Failure Models Indexed by Two Scales	231	(7)
Modelling the time-dependent covariate	231	(4)
Modelling the effect of the covariate on time to failure	235	(3)
Examples	238	(7)
The Poisson process	239	(1)
The doubly stochastic Poisson process	239	(1)
The compound Poisson process	240	(5)
A New Multiple Proof Loads Approach For Estimating Correlations	245	(14)
Richard Johnson
xsWenqing Lu
Introduction	245	(1)
Review of Current Designs for Estimating Correlation	246	(3)
A New Double Proof Load Design	249	(4)
Design, notation and the likelihood	249	(1)
The Fisher information for the double proof loading	250	(2)
Numerical results for the Fisher information	252	(1)
A Symmetric Version of the Double Proof Load Design	253	(1)
Asymptotic Results	254	(1)
An Example Using the Bivariate Weibull Distribution	255	(1)
Conclusion	256	(3)
Conditional and Partial Correlation For Graphical Uncertainty Models	259	(18)
Dorota Kurowicka
Roger Cooke
Introduction	259	(1)
Partial and Conditional Correlation	260	(2)
Trees, Vines and Copulas	262	(2)
Conditions for δ = 0	264	(2)
Copulae	266	(3)
Computing δ	269	(4)
Numerical Results	273	(2)
Conclusions	275	(2)
V Methods Common to Reliability and Survival Analysis	277	(46)
Semiparametric Methods of Time Scale Selection	279	(12)
Thierry Duchesne
Introduction	279	(1)
Modeling Failure	280	(2)
Semiparametric Inference	282	(3)
Semiparametric Model Assessment	285	(1)
Nonparametric Age Curves	286	(2)
Conclusion	288	(3)
Censored and Truncated Lifetime Data	291	(16)
Catherine Huber
Introduction	291	(2)
Examples of Censoring and Truncation	293	(1)
Censoring	293	(1)
Truncation	293	(1)
Regression Models: Independent Case	294	(3)
Cox model for independent individuals	294	(2)
Logistic discrete regression models	296	(1)
Accelerated models	296	(1)
Models for Clustered Data	297	(10)
Dependence as a nuisance	298	(2)
Models for estimating dependence	300	(7)
Tests for a Family of Survival Models Based on Extremes	307	(16)
Martin Crowder
The Situation	307	(2)
Distributions of Sample Extremes	309	(1)
A Test for v = 0	310	(2)
A Test for v = 1	312	(1)
A Test for k = 0	313	(1)
A Test for T = 0	313	(1)
Tests for v = T = 0	314	(1)
Tests for v - 1 = T = 0	315	(1)
Tests for k = T = 0	315	(2)
Applications	317	(6)
VI Software Reliability	323	(30)
Software Reliability Models - Past, Present and Future	325	(16)
M. Xie
Introduction	325	(1)
Some Important NHPP Software Reliability Models	326	(3)
General NHPP model formulation	326	(1)
Some models for finite failures	327	(1)
Some models for infinite failures	328	(1)
Further extension of the NHPP model	328	(1)
Markov Software Reliability Models	329	(2)
The Jelinski-Moranda (JM) model	329	(1)
A general formulation	330	(1)
The Shanthikumar general Markov model	331	(1)
Some Forecasting Approaches	331	(3)
Time series models and analysis	331	(1)
Regression models	332	(1)
Some Bayesian approaches	333	(1)
Future Distribution and Some Unsolved Problems	334	(7)
Further research on NHPP models	335	(1)
Interval estimation in software reliability analysis	335	(1)
Use of the models for decision making	335	(1)
Modelling and analysis of modular systems	336	(1)
Use of metrics in software reliability analysis	337	(4)
Dynamic Analysis of Failures in Repairable Systems and Software	341	(12)
Jerry F. Lawless
Introduction	341	(2)
Some Models and Associated Methodology	343	(2)
Some Specific Problems	345	(3)
Process monitoring and analysis	345	(1)
Prediction	346	(1)
Condition-based maintenance	347	(1)
Concluding Remarks	348	(5)
VI Statistical Inference	353	(96)
Precedence Test and Maximal Precedence Test	355	(24)
N. Balakrishnan
R. Frattina
Introduction	355	(1)
Precedence Test	356	(10)
Maximal Precedence Test	366	(11)
Possible Extensions	377	(2)
Hierarchical Bayesian Inference in Related Reliability Experiments	379	(12)
Duane L. Steffey
Francisco J. Samaniego
Hien Tran
Introduction	379	(2)
A Hierarchical Structure for Related Reliability Experiments	381	(1)
Approximate Posterior Inference	382	(2)
An Alternative: Linear Bayes Estimation	384	(1)
Comparison of Unpooled, Hierarchical Bayes, and Linear Bayes Estimators	384	(3)
Discussion	387	(4)
Tests for Equality of Intensities of Failures of a Repairable System Under Two Competing Risks	391	(14)
J.V. Deshpande
M. Mukhopadhyay
U.V. Naik-Nimbalkar
Introduction	391	(3)
Mathematical Preliminaries	394	(2)
Tests for Equality of Two Intensities	396	(2)
Consistency of the Tests	398	(2)
Asymptotic Relative Efficiency of the Tests for Equality	400	(2)
A.R.E. under alternative of proportional intensity	400	(1)
A.R.E. under alternative of proportionality of Log-cumulative-intensity	401	(1)
Analysis of Boring Machine Failure Data Using Competing Risk Theory	402	(3)
Semiparametric Estimation in Accelerated Life Testing	405	(14)
Vilijandas Bagdonavicius
Mikhail Nikulin
Models	405	(7)
Semiparametric Estimation	412	(2)
Properties of the Estimators	414	(5)
A Theoretical Framework for Accelerated Testing	419	(16)
Michael LuValle
Introduction	419	(4)
Thermal Demarcation Maps	423	(4)
Bounding Alternative Kinetics	427	(4)
Models 2, 3, and 4	427	(3)
Model 5	430	(1)
Open Questions and Conclusions	431	(4)
Unbiased Estimation in Reliability and Similar Problems	435	(14)
Mikhail Nikulin
Vassilly Voinov
Introduction	435	(2)
Univariate Continuous Probability Models	437	(4)
Multivariate Continuous Probability Models	441	(2)
Multivariate Discrete Probability Models	443	(6)
VIII Asymptotic Methods in Statistics	449	(51)
Prediction Under Association	451	(26)
George G. Roussas
Introduction, Notation, Assumption, and Statement of Main Result	451	(6)
Some Additional Notation and Auxiliary Results	457	(5)
Proof of Convergences (2.9) and (2.10)	462	(15)
Uniform Limit Laws for Kernel Density Estimators on Possibly Unbounded Intervals	477	(16)
Paul Deheuvels
Introduction and Main Results	477	(5)
Proofs	482	(11)
A Weak Convergence Result Relevant in Recurrent and Renewal Models	493	(7)
Edsel Pena
Robert L. Strawderman
Myles Hollander
Introduction	493	(1)
Mathematical Setting	494	(4)
Integral Transformed Processes	498	(2)
The Weak Convergence Result	500