Abstract
Based on a generalized cumulative damage approach with a stochastic process describing degradation, new accelerated life test models are presented in which both observed failures and degradation measures can be considered for parametric inference of system lifetime. Incorporating an accelerated test variable, we provide several new accelerated degradation models for failure based on the geometric Brownian motion or gamma process. It is shown that in most cases, our models for failure can be approximated closely by accelerated test versions of Birnbaum–Saunders and inverse Gaussian distributions. Estimation of model parameters and a model selection procedure are discussed, and two illustrative examples using real data for carbon-film resistors and fatigue crack size are presented.
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References
H. Akaike, “Information theory and an extension of the maximum likelihood principle,” in Proceedings of the Second International Symposium on Information Theory (B. N. Petrov and F. Czáki, eds.). Budapest: Akademiai Kiadó, pp. 267–281, 1973. Reprinted in Breakthroughs in Statistics, vol. 1(S. Kotz and N. L. Johnson, eds.). Berlin: Springer, pp. 610–624, 1993.
H. Akaike (1974) ArticleTitle“A new look at the statistical model identification” IEEE Transactions on Automatic Control 19 716–722 Occurrence Handle0314.62039 Occurrence Handle54 #11691
V. Bagdonavicius M. Nikulin (2000) ArticleTitle“Estimation in degradation models with explanatory variables” Lifetime Data Analysis 7 85–103 Occurrence Handle1819926
V. Bagdonavicius (2002) Nikulin, Accelerated Life Models, Modeling and Statistical Analysis Chapman & Hall /CRC Boca Raton, FL
G. K. Bhattacharyya A. Fries (1982) ArticleTitle“Fatigue failure models – Birnbaum–Saunders vs. inverse Gaussian” IEEE Transactions on Reliability 31 439–440
Z. W. Birnbaum S. C. Saunders (1969) ArticleTitle“A new family of life distributions” Journal of Applied Probability 6 319–327 Occurrence Handle40 #6707
M. Boulanger L. A. Escobar (1994) ArticleTitle“Experimental design for a class of accelerated degradation tests” Technometrics 36 260–272
K. P. Burnham D. R. Anderson (2002) Model Selection and Multi-Model Inference: A Practical Information-Theoretic Approach Springer New York
M. B. Carey R. H. Koenig (1991) ArticleTitle“Reliability assessment based on accelerated degradation: a case study” IEEE Transactions on Reliability 40 499–506 Occurrence Handle10.1109/24.106763
R. S. Chhikara J. L. Folks (1989) The Inverse Gaussian Distribution: Theory, Methodology, and Applications Marcel Dekker New York
D. R. Cox H. D. Miller (1965) The Theory of Stochastic Processes John Wiley & Sons New York
K. Doksum S.- L. T. Normand (1995) ArticleTitle“Gaussian models for degradation processes – Part I: Methods for the analysis of biomarker data” Lifetime Data Analysis 1 135–144 Occurrence Handle10.1007/BF00985763 Occurrence Handle96h:62151
S. D. Durham W. J. Padgett (1997) ArticleTitle“A cumulative damage model for system failure with application to carbon fibers and composites” Technometrics 39 34–44
M. Hamada (1995) “Analysis of experiments for reliability improvement and robust reliability” N. Balakrishnan (Eds) Recent Advances in Life Testing and Reliability CRC Press Boca Raton, FL
S. J. Hudak SuffixJr A. Saxena R. J. Bucci R. C. Malcolm (1978) Development of Standard Methods of Testing and Analyzing Fatigue Crack Growth Rate Data Technical Report AFML-TR-78-40 Westinghouse R & D Center
J. Jacod A. N. Shiryaev (1987) Limit Theorems for Stochastic Processes Springer-Verlag New York
J. Lawless M. Crowder (2004) ArticleTitle“Covariates and random effects in a gamma process model with application to degradation and failure” Lifetime Data Analysis 10 213–227 Occurrence Handle10.1023/B:LIDA.0000036389.14073.dd Occurrence Handle2086957
C. J. Lu W. Q. Meeker (1993) ArticleTitle“Using degradation measures to estimate a time-to-failure distribution” Technometrics 35 161–174 Occurrence Handle1225093
J. Lu, “Degradation Processes and Related Reliability Models,” Ph.D. thesis, McGill University, 1995.
N. R. Mann R. E. Schafer N. D. Singpurwalla (1974) Methods for Statistical Analysis of Reliability and Life Data John Wiley & Sons New York
W. Q. Meeker L. A. Escobar (1998) Statistical Methods for Reliability Data John Wiley & Sons New York
W. Q. Meeker L. A. Escobar C. J. Lu (1998) ArticleTitle“Accelerated degradation tests: modeling and analysis” Technometrics 40 89–99
W. Nelson (1990) Accelerated Testing: Statistical Models, Test Plans, and Data Analyses John Wiley & Sons New York
A. Onar W. J. Padgett (2000) ArticleTitle“Inverse Gaussian accelerated test models based on cumulative damage” Journal of Statistical Computation and Simulation 66 233–247 Occurrence Handle1807537
W. J. Owen W. J. Padgett (1999) ArticleTitle“Accelerated test models for system strength based on Birnbaum–Saunders distributions” Lifetime Data Analysis 5 133–147 Occurrence Handle10.1023/A:1009649428243 Occurrence Handle1750394
W. J. Padgett M. A. Tomlinson (2004) ArticleTitle“Inference from accelerated degradation and failure data based on Gaussian process models” Lifetime Data Analysis 10 191–206 Occurrence Handle10.1023/B:LIDA.0000030203.49001.b6 Occurrence Handle2081721
C. Park W. J. Padgett (2005) ArticleTitle“New cumulative damage models for failure using stochastic processes as initial damage,” IEEE Transactions on Reliability 54 530–540
L. I. Pettit K. D. S. Young (1999) ArticleTitle“Bayesian analysis for inverse Gaussian lifetime data with measures of degradation” Journal of Statistical Computation and Simulation 63 217–234 Occurrence Handle1703821
N. U. Prabhu (1965) Stochastic Processes: Basic Theory and Its Applications Macmillan Company New York
Y. Shiomi T. Yanagisawa (1979,) ArticleTitle“On distribution parameter during accelerated life test for a carbon film resistor” Bulletin of Electrotechnical Laboratory 43 330–345
K. Suzuki K. Maki S. Yokogawa (1993) “An analysis of degradation data of a carbon film and the properties of the estimators” K. Matusita M. L. Puri T. Hayakawa (Eds) Proceedings of the Third Pacific Area Statistical Conference Zeist The Netherlands 501–511
G. A. Whitmore (1995) ArticleTitle“Estimating degradation by a wiener diffusion process subject to measurement error” Lifetime Data Analysis 1 307–319 Occurrence Handle10.1007/BF00985762 Occurrence Handle0960.62540
G. A. Whitmore M. J. Crowder J.F. Lawless (1998) ArticleTitle“Failure inference from a marker process based on a bivariate Wiener model” Lifetime Data Analysis 4 229–251 Occurrence Handle10.1023/A:1009617814586
G. A. Whitmore F. Schenkelberg (1997) ArticleTitle“Modelling accelerated degradation data using wiener diffusion with a scale transformation” Lifetime Data Analysis 3 27–45 Occurrence Handle10.1023/A:1009664101413
M. B. Wilk R. Gnanadesikan (1968) ArticleTitle“Probability plotting methods for the analysis of data” Biometrika 55 1–17
T. Yanagisawa (1997) ArticleTitle“Estimation of the degradation of amorphous silicon cells” Microelectronics and Reliability 37 549–554
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Park, C., Padgett, W.J. Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes. Lifetime Data Anal 11, 511–527 (2005). https://doi.org/10.1007/s10985-005-5237-8
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DOI: https://doi.org/10.1007/s10985-005-5237-8