Abstract
In this chapter we describe a simple degradation model based on the Wiener process. A failure occurs when the degradation reaches a given level for the first time. In this case, the time to failure is inverse Gaussian distributed. The parameters of the lifetime distribution can be estimated from observation of degradation only, from observation of failures, or from observation of both degradation increments and failure times. In the chapter, statistical methods for estimating the parameters of degradation processes for different data structures are developed and compared.
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References
Barndorff-Nielsen, O. E., Blaesild, P. (1986). A note on the calculation of Bartlett adjustments, Journal of Royal Statististical Society Series B 48, 353–358.
Cordeiro, G. M. (1987). On the corrections to the likelihood ratio statistics, Biometrika 74, 265–274.
Cox, D. R. and Reid, N. (1987). Parameter orthogonality and approximate conditional inference, Journal of Royal Statististical Society Series B 49, 1–39.
Doksum, K. A. and Normand, S. T. (1996) Models for degradation processes and event times based on gaussian processes. In: Jewell, N. P. et al. (eds), Lifetime Data: Models in Reliability and Survival Analysis. Netherlands: Kluwer academic publishers, 85–91.
Kahle, W. (1994). Simultaneous confidence regions for the arameters of damage processes. Statistical Papers 35, 27–41.
Kahle, W. Lehmann, A. (1998) Parameter estimation in damage processes: Dependent observations of damage increments and first passage time, In: Kahle, W. et al. (eds), Advances in Stochastic Models for Reliability, Quality and Safety. Boston: Birkhauser, pp 139–152.
Lu, C. J. and Meeker, W. Q. (1993) Using degradation measures to estimate a time-to-failure distribution, Technometrics, 35, 161–174.
Wentzell, A. D. (1979). Theorie zufälliger Prozesse. Berlin: Akademie-Verlag.
Whitmore, G. A. (1995) Estimating degradation by a Wiener diffusion process subject to measurement error, Lifetime Data Analysis, 1, 307–319.
Whitmore, G. A., and Schenkelberg, F. (1997) Modelling accelerated degradation data using Wiener diffusion with a time scale transformation, Lifetime Data Analysis, 3, 27–45.
Whitmore, G. A., Crowder, M. J., and Lawless, J. F. (1998) Failure inference from a marker process based on a bivariate Wiener model, Lifetime Data Analysis, 4, 229–251.
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Kahle, W., Lehmann, A. (2010). The Wiener Process as a Degradation Model: Modeling and Parameter Estimation. In: Nikulin, M., Limnios, N., Balakrishnan, N., Kahle, W., Huber-Carol, C. (eds) Advances in Degradation Modeling. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4924-1_9
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DOI: https://doi.org/10.1007/978-0-8176-4924-1_9
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