Rykov / Balakrishnan / Nikulin Mathematical and Statistical Models and Methods in Reliability

Applications to Medicine, Finance, and Quality Control
1. Auflage 2010
ISBN: 978-0-8176-4971-5
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 457 Seiten

Reihe: Mathematics and Statistics

The book is a selection of invited chapters, all of which deal with various aspects of mathematical and statistical models and methods in reliability. Written by renowned experts in the field of reliability, the contributions cover a wide range of applications, reflecting recent developments in areas such as survival analysis, aging, lifetime data analysis, artificial intelligence, medicine, carcinogenesis studies, nuclear power, financial modeling, aircraft engineering, quality control, and transportation. Mathematical and Statistical Models and Methods in Reliability is an excellent reference text for researchers and practitioners in applied probability and statistics, industrial statistics, engineering, medicine, finance, transportation, the oil and gas industry, and artificial intelligence.

Autoren/Hrsg.

Rykov, V.V.

Balakrishnan, N.

Nikulin, M.S.

Rykov, V. V.

Nikulin, M. S.

Inhaltsverzeichnis

1;Contents;6
2;Preface;16
3;List of Contributors;18
4;List of Tables;22
5;List of Figures;24
6;Reliability Models, Methods, and Optimization;28
6.1;1 Reliability of Semi-Markov Systems with Asymptotic Merging Phase Space;29
6.1.1;1.1 Introduction;29
6.1.2;1.2 Reliability of the Renewal System;30
6.1.3;1.3 Absorbing Time of Semi-Markov Process;33
6.1.4;1.4 Reward Functional in the Semi-Markov Environment;37
6.1.5;1.5 Dynamic Reward Functional;38
6.1.6;1.6 Fluctuations of the Reward Functional;39
6.1.7;1.7 Heuristic Phase Merging;39
6.1.8;1.8 Stationary Phase Merging Scheme;40
6.1.9;References;43
6.2;2 Nonlinearly Perturbed Stochastic Processes and Systems;45
6.2.1;2.1 Introduction;45
6.2.2;2.2 Nonlinearly Perturbed Renewal Equation;47
6.2.3;2.3 Nonlinearly Perturbed Regenerative Processes;53
6.2.4;2.4 Nonlinearly Perturbed Semi-Markov Processes;56
6.2.5;2.5 Nonlinearly Perturbed Stochastic Systems;57
6.2.6;2.6 Nonlinearly Perturbed Risk Processes;58
6.2.7;2.7 Conclusion;59
6.2.8;References;60
6.3;3 On a Copula for Failure Times of System Elements;64
6.3.1;3.1 Introduction;64
6.3.2;3.2 Multidimensional Distribution;66
6.3.3;3.3 Different Properties;68
6.3.4;3.4 Copula;71
6.3.5;3.5 Example;72
6.3.6;3.6 Conclusion;73
6.3.7;3.7 Appendix;74
6.3.8;References;75
6.4;4 On One Method of Reliability Coefficients Calculation for Objects in Non- Homogeneous Event Flows;76
6.4.1;4.1 Introduction;76
6.4.2;4.2 General Concepts;77
6.4.3;4.3 Review of Models Taking into Account Non-Homogeneity;77
6.4.4;4.4 Normalizing Flow Function Model;78
6.4.5;4.5 Leading Flow Function and Failure Flow Parameter;80
6.4.6;4.6 Getting of Distribution Law for the Operability Cycle;81
6.4.7;4.7 Getting of Resource Reliability Characteristics;83
6.4.8;4.8 Availability Function Calculation;85
6.4.9;4.9 Numerical Computation of Availability Function;89
6.4.10;References;90
6.4.11;Appendix;91
6.5;5 A New Approach to Maintenance Optimization by Modeling Intensity Control;93
6.5.1;5.1 Introduction;94
6.5.2;5.2 Model;95
6.5.3;5.3 Modelling Intensity Control Given Partial Information;100
6.5.4;5.4 Numerical Example;104
6.5.5;References;108
6.6;6 Longitudinal Latent Markov Processes Observable Through an Invariant Rasch Model;110
6.6.1;6.1 Introduction;110
6.6.2;6.2 Description of the Latent Process;111
6.6.3;6.3 Application to Markov Latent Processes;114
6.6.4;6.4 Estimation;118
6.6.5;6.5 Estimation of Rasch Latent Markov Processes;120
6.6.6;References;122
6.7;7 Dynamics of Dependence Properties for Lifetimes Influenced by Unobservable Environmental Factors;124
6.7.1;7.1 Introduction;124
6.7.2;7.2 Some Basic Facts;127
6.7.3;7.3 Monotonicity Properties of Survival Copulas;129
6.7.4;7.4 Concluding Remarks;133
6.7.5;References;134
6.7.6;Appendix;134
6.8;8 On Alternative of Choice for a Prophylaxis Problem;136
6.8.1;8.1 Renewal System;136
6.8.2;8.2 Optimization Problem;138
6.8.3;8.3 Alternative of Choice for Twice Regenerative System;139
6.8.4;References;145
6.9;9 Optimal Incomplete Maintenance for Systems with Discrete Time- to- Failure Distribution;146
6.9.1;9.1 Introduction;146
6.9.2;9.2 Discrete Lifetime Distributions and Their Failure Rate;146
6.9.3;9.3 Kijima Type Repairs;148
6.9.4;9.4 Optimal Maintenance as Time Scale Transformation;149
6.9.5;9.5 Simulation of Failure–Repair Processes;151
6.9.6;9.6 Cost Optimal Maintenance;153
6.9.7;9.7 Conclusion;155
6.9.8;References;155
6.10;10 A Gini-Type Index for Aging/Rejuvenating Objects;156
6.10.1;Acronyms;156
6.10.2;10.1 Introduction;156
6.10.3;10.2 GT Coefficient for Repairable Systems;158
6.10.4;10.3 GT Coefficient for Non-repairable Systems (Components);160
6.10.5;10.4 Conclusions;162
6.10.6;References;163
6.11;11 Redundancy Analysis for Multi-state System: Reliability and Financial Assessment;164
6.11.1;11.1 Introduction;164
6.11.2;11.2 Problem Formulation;166
6.11.3;11.3 Model Descriptions;168
6.11.4;11.4 Algorithm of UGF Computation for Entire MSS;176
6.11.5;11.5 Reliability Measures Computation for Entire MSS;177
6.11.6;11.6 Illustrative Example;177
6.11.7;11.7 Conclusions;179
6.11.8;References;179
6.12;12 On the Reliability Modeling of Hierarchical Systems;181
6.12.1;12.1 Introduction and Motivation;181
6.12.2;12.2 General Model;182
6.12.3;12.3 Investigation of the Reliability of a General Model;183
6.12.4;12.4 The Subsystems Behavior Investigation;184
6.12.5;12.5 An Example;193
6.12.6;12.6 Further Problems;194
6.12.7;References;195
7;Statistical Methods in Reliability;197
7.1;13 Parametric Estimation of Redundant System Reliability From Censored Data;198
7.1.1;13.1 Introduction;198
7.1.2;13.2 Parametric Point Estimators of the c.d.f. Kj( t);199
7.1.3;13.3 Asymptotic Confidence Intervals for Kj( t);200
7.1.4;References;212
7.2;14 Assessing Accuracy of Statistical Inferences by Resamplings;213
7.2.1;14.1 Introduction;213
7.2.2;14.2 Linear Heteroscedastic Regression Model;215
7.2.3;14.3 Resampling from Estimates of Residuals;217
7.2.4;14.4 OLS-Estimators of Regression Coefficients and Their Accuracies;219
7.2.5;14.5 Conclusion;221
7.2.6;References;223
7.2.7;Appendix;224
7.3;15 Change Point Estimation in Regression Models with Fixed Design;227
7.3.1;15.1 Introduction;227
7.3.2;15.2 The Jump Case;230
7.3.3;15.3 The Continuous Case;236
7.3.4;References;239
7.3.5;Appendix;239
7.4;16 A Model for Field Failure Prediction Using Dynamic Environmental Data;242
7.4.1;16.1 Introduction;242
7.4.2;16.2 Data and Model;244
7.4.3;16.3 Parameter Estimation;247
7.4.4;16.4 Predictions;248
7.4.5;16.5 Calibration of Prediction Intervals;249
7.4.6;16.6 Conclusions and Areas for Future Research;250
7.4.7;References;251
7.5;17 Efficient Regression Estimation Under General Censoring and Truncation;253
7.5.1;17.1 Framework;253
7.5.2;17.2 Data Description;254
7.5.3;17.3 The Semi-Parametric Model;255
7.5.4;17.4 Inference About ß;258
7.5.5;17.5 Perpectives and References;259
7.5.6;References;259
7.6;18 On Generalized Tests of Fit for Multinomial Populations;260
7.6.1;18.1 Introduction;260
7.6.2;18.2 The F-Family of Test Statistics;263
7.6.3;18.3 Empirical Distribution and Power Simulations;264
7.6.4;18.4 Conclusions;269
7.6.5;References;270
7.7;19 Modeling and Scaling of Categorical Data;272
7.7.1;19.1 Introduction;272
7.7.2;19.2 Basic Model;273
7.7.3;19.3 Criteria for Scaling;277
7.7.4;19.4 Calculation of Most Separating Scales;279
7.7.5;References;281
7.8;20 Nonparametric Estimation and Testing the Effect of Covariates in Accelerated Life Time Models Under Censoring;282
7.8.1;20.1 Introduction;282
7.8.2;20.2 Nonparametric Estimation of the Regression Function Under Censoring;284
7.8.3;20.3 Properties of the Nonparametric Regression Estimator;286
7.8.4;20.4 Formulation of the Test Procedure;291
7.8.5;References;293
7.9;21 Nonparametric Estimation of Time Trend for Repairable Systems Data;294
7.9.1;21.1 Introduction;294
7.9.2;21.2 Definitions and Preliminaries;295
7.9.3;21.3 A Motivating Example;297
7.9.4;21.4 Nonparametric Estimation of a Monotone .(·);298
7.9.5;21.5 Kernel-Estimators for .(t) in the General Case;299
7.9.6;21.6 Concluding Remarks;303
7.9.7;References;305
7.10;22 Confidence Region for Distribution Function from Censored Data;306
7.10.1;22.1 Introduction;306
7.10.2;22.2 Testing of the Homogeneity Hypothesis;311
7.10.3;References;312
7.11;23 Empirical Estimate with Uniformly Minimal d-Risk for Bernoulli Trials Success Probability;313
7.11.1;23.1 Introduction;313
7.11.2;23.2 An Estimate with Uniformly Minimal d-Risk;314
7.11.3;23.3 Empirical Estimate Based on the Modification of the Historical Data;316
7.11.4;23.4 Empirical Estimate Based on the Invariance of EUMD for the Scalar Parameter of a Discrete Exponential Family;319
7.11.5;23.5 Accuracy Properties Investigation for the Estimates Based on the Statistical Modeling;321
7.11.6;References;322
7.12;24 Estimation of Archival Lifetime Distribution for Writable Optical Disks from Accelerated Testings;323
7.12.1;24.1 Introduction;323
7.12.2;24.2 Models;323
7.12.3;24.3 Estimation of Parameters and Percentiles;325
7.12.4;24.4 An Example;327
7.12.5;24.5 Concluding Remarks;328
7.12.6;References;328
8;Applications;330
8.1;25 Ages in Reliability and Bio Systems, Interpretations, Control, and Applications;331
8.1.1;25.1 Introduction;331
8.1.2;25.2 Main Definitions and Equivalent Representations;332
8.1.3;25.3 Empirical Distribution Functions and Empirically Equivalent Ages;335
8.1.4;25.4 Equivalent Ages Have Equivalent/Equal Accumulated Stress: Comparing Ages of Processes;336
8.1.5;25.5 Multidimensional Life Time Equivalence;343
8.1.6;25.6 Conclusions;347
8.1.7;References;347
8.2;26 Shocks in Mixed Populations;349
8.2.1;26.1 Introduction;349
8.2.2;26.2 Two Ordered Subpopulations;350
8.2.3;26.3 Continuous Mixtures;353
8.2.4;26.4 Concluding Remarks;357
8.2.5;References;357
8.3;27 Bayesian Estimation of Degradation Model Defined by a Wiener Process;359
8.3.1;27.1 Introduction;359
8.3.2;27.2 Reliability Testing;360
8.3.3;27.3 Bayesian Estimation of Wiener Process Parameters;362
8.3.4;27.4 Estimation of Prior Mean of Wiener Process by FEM Method: Application to Brake Disc Wear;366
8.3.5;27.5 Results;368
8.3.6;27.6 Conclusion;370
8.3.7;References;371
8.4;28 Benefits of Threshold Regression: A Case-Study Comparison with Cox Proportional Hazards Regression;372
8.4.1;28.1 Introduction;372
8.4.2;28.2 First-Hitting Time (FHT) and Threshold Regression (TR) Model;373
8.4.3;28.3 Threshold Regression (TR) Investigations of Lung Cancer;375
8.4.4;28.4 Comparisons of Results Obtained from the Cox PH Model;378
8.4.5;References;382
8.5;29 Optimal Stopping and Reselling of European Options;384
8.5.1;29.1 Introduction;384
8.5.2;29.2 Formulation of the Reselling Problem;385
8.5.3;29.3 Convergence of Rewards for Multivariate Markov Price Processes;387
8.5.4;29.4 Binomial–Trinomial Approximations for Reselling Model;391
8.5.5;29.5 Numerical Examples;398
8.5.6;References;400
8.6;30 Bayesian Modeling of Health State Preferences;403
8.6.1;30.1 Introduction;403
8.6.2;30.2 Bayesian Modeling of Attribute Utilities;405
8.6.3;30.3 Modeling Attribute Weights;407
8.6.4;30.4 Bayesian Evaluation of Health States;409
8.6.5;30.5 Concluding Remarks;410
8.6.6;References;410
8.7;31 Information Measures in Biostatistics and Reliability Engineering;412
8.7.1;31.1 Introduction;412
8.7.2;31.2 Modelling in Biomedicine and Reliability;413
8.7.3;31.3 Applications;420
8.7.4;References;422
9;Reliability Computer Tools;425
9.1;32 Software System for Simulation and Research of Probabilistic Regularities and Statistical Data Analysis in Reliability and Quality Control;426
9.1.1;32.1 Introduction;426
9.1.2;32.2 The Investigation of Parameter Estimates Properties;427
9.1.3;32.3 The Investigation of Nonparametric Goodness-of-Fit Test Statistic Distributions;427
9.1.4;32.4 The Investigation of Statistic Distributions and the Power of .2 Tests;429
9.1.5;32.5 The Comparative Analysis of the Power of Goodness- of- Fit Tests;432
9.1.6;32.6 The Investigation of Statistic Distributions and the Power of Normality Tests;433
9.1.7;32.7 The Investigation of Homogeneity Test Statistic Distributions;433
9.1.8;32.8 The Investigation of Statistic Distributions and the Power of Tests for Homogeneity of Means;434
9.1.9;32.9 The Investigation of Statistic Distributions and the Power of Tests for Homogeneity of Variances;435
9.1.10;32.10 Conclusion;437
9.1.11;References;438
9.2;33 Inverse Gaussian Model and Its Applications in Reliability and Survival Analysis;442
9.2.1;33.1 Introduction;442
9.2.2;33.2 The Family of the Inverse Gaussian Distributions;443
9.2.3;33.3 Goodness-of-Fit Tests for the Family of IGD;446
9.2.4;33.4 About Testing Hypotheses for the Inverse Gaussian Distribution;448
9.2.5;33.5 Chi-Squared Goodness-of-fit Test for the Family of IGD in Case of Censored Data;454
9.2.6;33.6 Models with Covariates Based on the Family of IGD;456
9.2.7;References;461
10;Index;463