Parameter estimation from load-sharing system data using the expectation–maximization algorithm

References

• Albert , J.R. G. and Baxter , L. A. 1995 . Applications of the EM algorithm to the analysis of life length data . Applied Statistics , 44 : 323 – 341 .
   Web of Science ®Google Scholar
• Amari , S. V. and Bergman , R. 2008 . “ Reliability analysis of k-out-of-n load-sharing systems ” . In Proceedings of the Annual Reliability and Maintainability Symposium 440 – 445 .
   Google Scholar
• Amari , S.V. , Misra , K.B. and Pham , H. 2008 . “ Tampered failure rate load-sharing systems: status and perspectives ” . In Handbook of Performability Engineering , Edited by: Misra , K.B. 291 – 308 . London : Springer .
   Google Scholar
• Amari , S. V. , Mohan , K. , Cline , B. and Xing , L. 2009 . Expectation–maximization algorithm for failure analysis using incomplete warranty data . International Journal of Performability Engineering , 5 : 403 – 417 .
   Google Scholar
• Amari , S. V. and Pham , H. 2009 . “ Reliability analysis of dynamic fiber bundle models ” . In Proceedings of the Annual Reliability and Maintainability Symposium 26 – 29 .
   Google Scholar
• Balakrishnan , N. , Beutner , E. and Kamps , U. 2008 . Order restricted inference for sequential k-out-of-n systems . Journal of Multivariate Analysis , 99 : 1489 – 1502 .
   Web of Science ®Google Scholar
• Balakrishnan , N. and Mi , J. 2003 . Existence and uniqueness of the MLEs for normal distribution based on general progressively Type-II censored samples . Statistics & Probability Letters , 64 : 407 – 414 .
   Web of Science ®Google Scholar
• Barlow , R. E. 1998 . Engineering Reliability , Philadelphia , PA : ASA-SIAM .
   Google Scholar
• Belzunce , F. , Mercader , J. A. and Ruiz , J. M. 2003 . Multivariate aging properties of epoch times of nonhomogeneous processes . Journal of Multivariate Analysis , 84 : 335 – 350 .
   Web of Science ®Google Scholar
• Bhattacharyya , G. K. and Soejoeti , Z. 1989 . A tampered failure rate model for step-stress accelerated life test . Communications in Statistics: Theory and Methods , 18 : 1627 – 1643 .
   Web of Science ®Google Scholar
• Birnbaum , Z. W. and Saunders , S. C. 1958 . A statistical model for life length of material . Journal of the American Statistical Association , 53 : 151 – 160 .
   Web of Science ®Google Scholar
• Broyden , C. G. 1972 . Quasi-Newton Methods in Numerical Methods for Unconstrained Optimization , New York , NY : Academic Press .
   Google Scholar
• Burkschat , M. , Kamps , U. and Kateri , M. 2010 . Sequential order statistics with an order statistic prior . Journal of Multivariate Analysis , 101 : 1826 – 1836 .
   Web of Science ®Google Scholar
• Casella , G. and Berger , R.L. 2002 . Statistical Inference , Duxbury, Pacific Grove, CA : second edition .
   Google Scholar
• Coleman , B. 1957a . Time dependence of mechanical breakdown in bundles of fibers I. Constant total load . Journal of Applied Physics , 28 : 1058 – 1064 .
   Web of Science ®Google Scholar
• Coleman , B. 1957b . Time dependence of mechanical breakdown in bundles of fibers II. The infinite ideal bundle under linearly increasing loads . Journal of Applied Physics , 28 : 1065 – 1067 .
   Web of Science ®Google Scholar
• Cox , D. R. 1972 . Regression models and life tables . Journal of the Royal Statistical Society Series B , 34 : 187 – 220 .
   Web of Science ®Google Scholar
• Cramer , E. and Kamps , U. 2001 . “ Sequential k-out-of-n systems ” . In Handbook of Statistics , Edited by: Rao , Balakrishnan N., . 301 – 372 . Amsterdam, C.R. , , The Netherlands : Elsevier Science .
   Google Scholar
• Crow , E. L. and Shimizu , K. 1988 . Lognormal Distributions: Theory and Applications , New York , NY : Marcel Dekker .
   Google Scholar
• Daniels , H. E. 1945 . The statistical theory of the strength of bundles of threads I . Proceedings of the Royal Society of London, Series A , 83 : 405 – 435 .
   Google Scholar
• David , H. A. and Moeschberger , M. L. 1978 . The Theory of Competing Risks , New York , NY : Macmillan .
   Google Scholar
• Dempster , A.P. , Laird , N.M. and Rubin , D.B. 1977 . “ Maximum likelihood from incomplete data via the EM algorithm ” . In Journal of the Royal Statistical Society , 1 – 22 . Series, B, 9 .
   Google Scholar
• Durham , S. D. , Lynch , J. D. and Padgett , W. J. 1989 . A theoretical justification for an increasing average failure rate strength distribution in fibrous composites . Naval Research Logistics , 36 : 655 – 661 .
   Web of Science ®Google Scholar
• Durham , S. D. , Lynch , J. D. , Padgett , W. J. , Horan , T. J. , Owen , W. J. and Surles , J. 1997 . Localized load-sharing rules and Markov–Weibull fibers: a comparision of microcomposite failure data with Monte Carlo simulations . Journal of Composite Materials , 31 : 1856 – 1882 .
   Web of Science ®Google Scholar
• Fletcher , R. 1970 . A new approach to variable metric algorithms . Computer Journal , 13 : 317 – 322 .
   Web of Science ®Google Scholar
• Fletcher , R. 2000 . Practical Methods of Optimization , New York , NY : Wiley .
   Google Scholar
• Goldfarb , D. 1970 . A family of variable metric updates derived by variational means . Mathematics of Computation , 24 : 23 – 26 .
   Web of Science ®Google Scholar
• Goldthwaite , L. 1961 . “ Failure rate study for the lognormal lifetime model ” . In Proceedings of the Seventh National Symposium on Reliability and Quality Control 208 – 213 .
   Google Scholar
• Harlow , D. G. and Phoenix , S. L. 1978 . The chain-of-bundles probability model for the strength of fibrous materials I. Analysis and conjectures . Journal of Composite Materials , 12 : 195 – 214 .
   Web of Science ®Google Scholar
• Harlow , D. G. and Phoenix , S. L. 1982 . Probability distributions for the strength of fibrous materials under local load sharing I. Two-level failure and edge effects . Advances in Applied Probability , 14 : 68 – 94 .
   Web of Science ®Google Scholar
• Heitjan , D. F. and Rubin , D. B. 1990 . Inference from coarse data via multiple imputation with application to age heaping . Journal of the American Statistical Association , 85 : 304 – 314 .
   Web of Science ®Google Scholar
• Heitjan , D. F. and Rubin , D. B. 1991 . Ignorability and coarse data . Annals of Statistics , 19 : 2244 – 2253 .
   Web of Science ®Google Scholar
• Hollander , M. and Pe na , E. A. 1995 . Dynamic reliability models with conditional proportional hazards . Lifetime Data Analysis , 1 : 377 – 401 .
   PubMedGoogle Scholar
• Ihaka , R. and Gentleman , R. 1996 . R: A language for data analysis and graphics . Journal of Computational and Graphical Statistics , 5 : 299 – 314 .
   Google Scholar
• Kalbfleisch , J. D. and Prentice , R. L. 2002 . The Statistical Analysis of Failure Time Data , New York , NY : Wiley .
   Google Scholar
• Kamps , U. 1995 . A concept of generalized order statistics . Journal of Statistical Planning and Inference , 48 : 1 – 23 .
   Web of Science ®Google Scholar
• Kim , H. and Kvam , P. H. 2004 . Reliability estimation based on system data with an unknown load share rule . Lifetime Data Analysis , 10 : 83 – 94 .
   PubMed Web of Science ®Google Scholar
• Kim , J. S. and Yum , B.-J . 2008 . Selection between Weibull and lognormal distributions: a comparative simulation study . Computational Statistics & Data Analysis , 53 : 477 – 485 .
   Web of Science ®Google Scholar
• Kvam , P. H. and Pe na , E. A. 2005 . Estimating load-sharing properties in a dynamic reliability system . Journal of the American Statistical Association , 100 : 262 – 272 .
   PubMed Web of Science ®Google Scholar
• Lee , S. , Durham , S. and Lynch , J. 1995 . On the calculation of the reliability of general load sharing system . Journal of Applied Probability , 32 : 777 – 792 .
   Web of Science ®Google Scholar
• Lee , S. B. and Park , C. 2006 . Development of robust design optimization using incomplete data . Computers & Industrial Engineering , 50 : 345 – 356 .
   Web of Science ®Google Scholar
• Leemis , L. M. 1995 . Reliability , Englewood Cliffs , NJ : Prentice-Hall .
   Google Scholar
• Little , R.J. A. and Rubin , D. B. 2002 . Statistical Analysis with Missing Data , New York , Wiley NY : second edition .
   Google Scholar
• Mann , N. R. , Schafer , R. E. and Singpurwalla , N. D. 1974 . Methods for Statistical Analysis of Reliability and Life Data , New York , NY : Wiley .
   Google Scholar
• Martz , H. F. and Waller , R. A. 1982 . Bayesian Reliability Analysis , New York , NY : Wiley .
   Google Scholar
• MATLAB . 2010 . MATLAB version 7.10.0, R2010a , Natick, MA : The MathWorks. Inc. .
   Google Scholar
• McLachlan , G. J. and Krishnan , T. 1997 . The EM Algorithm and Extensions , New York , NY : Wiley .
   Google Scholar
• McLachlan , G. J. and Peel , D. 2000 . Finite Mixture Models , New York , NY : Wiley .
   Google Scholar
• Mettas , A. and Vassiliou , P. 2004 . “ Application of quantitative accelerated life models on load sharing redundancy ” . In Proceedings of Annual Reliability and Maintainability Symposium 293 – 296 .
   Google Scholar
• Minitab. 2010 . Minitab Statistical Software , Minitab : release 16 .
   Google Scholar
• Nelson , W. 1980 . Accelerated life testing step-stress models and data analysis . IEEE Transactions on Reliability , R-29 : 103 – 108 .
   Web of Science ®Google Scholar
• Ortega , J. M. and Rheinboldt , W. C. 1970 . Iterative Solutions on Nonlinear Equations in Several Variables , New York , NY : Academic Press .
   Google Scholar
• Park , C. 2005 . Parameter estimation of incomplete data in competing risks using the EM algorithm . IEEE Transactions on Reliability , 54 : 282 – 290 .
   Web of Science ®Google Scholar
• Park , C. 2006 . “ A quantile implementation of the EM algorithm and applications to parameter estimation with interval data. ” . In Technical Report TR2006-05-CP, Department of Mathematical Sciences , Clemson, SC : Clemson University .
   Google Scholar
• Park , C. 2010 . Parameter estimation for reliability of load sharing systems . IIE Transactions , 42 : 753 – 765 .
   Web of Science ®Google Scholar
• Park , C. and Lee , S.B. 2003 . Parameter estimation from censored samples using the expectation-maximization algorithm. , Clemson, SC : Technical Report TR2003-05-PL, Department of Mathematical Sciences, Clemson University .
   Google Scholar
• Park , C. and Padgett , W. J. 2005 . New cumulative damage models for failure using stochastic processes as initial damage . IEEE Transactions on Reliability , 54 : 530 – 540 .
   Web of Science ®Google Scholar
• Park , C. and Padgett , W. J. 2006a . Analysis of strength distributions of multi-modal failures using the EM algorithm . Journal of Statistical Computation and Simulation , 76 : 619 – 636 .
   Web of Science ®Google Scholar
• Park , C. and Padgett , W. J. 2006b . Stochastic degradation models with several accelerating variables . IEEE Transactions on Reliability , 55 : 379 – 390 .
   Web of Science ®Google Scholar
• Phoenix , S. L. 1978 . The asymptotic time to failure of a mechanical system of parallel members . SIAM Journal of Applied Mathematics , 34 : 227 – 246 .
   Web of Science ®Google Scholar
• Pike , M. C. 1966 . A method of analysis of a certain class of experiments in carcinogenesis . Biometrics , 22 : 142 – 161 .
   PubMed Web of Science ®Google Scholar
• R Development Core Team. 2011 . R: A Language and Environment for Statistical Computing , Vienna, Austria : R Foundation for Statistical Computing .
   Google Scholar
• Robert , C. P. and Casella , G. 1999 . Monte Carlo Statistical Methods , New York : Springer .
   Google Scholar
• Rosen , B. W. 1964 . Tensile failure of fibrous composites . AIAA Journal , 2 : 1985 – 1991 .
   Web of Science ®Google Scholar
• Ross , S. M. 1984 . A model in which component failure rates depend on working set . Naval Research Logistics Quarterly , 31 : 297 – 300 .
   Google Scholar
• Schafer , J. L. 1997 . Analysis of Incomplete Multivariate Data , Boca Raton, FL : Chapman & Hall .
   Google Scholar
• Schechner , Z. 1984 . A load-sharing model: the linear breakdown rule . Naval Research Logistics Quarterly , 31 : 137 – 144 .
   Google Scholar
• Shanno , D. F. 1970 . Conditioning of quasi-Newton methods for function minimization . Mathematics of Computation , 24 : 647 – 656 .
   Web of Science ®Google Scholar
• Tanner , M. A. 1996 . Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions , New York : Springer-Verlag .
   Google Scholar
• Tiwari , R. , Kulasekera , K. B. and Park , C. 2006 . Nonparametric tests for cause specific hazard rates with censored data in competing risks among several groups . Journal of Statistical Planning and Inference , 136 : 1718 – 1745 .
   Web of Science ®Google Scholar