Estimating the Parameters of Degradation Models when Error Terms are Autocorrelated
DOI:
https://doi.org/10.17713/ajs.v40i3.210Abstract
The need of autocorrelation models for degradation data comes from the facts that the degradation measurements are often correlated, since such measurements are taken over time. Time series can exhibit autocorrelation caused by modeling error or cyclic changes in ambient conditions in the measurement errors or in degradation process itself. Generally, autocorrelation becomes stronger when the times between measurements are relativelyshort and becomes less noticeable when the times between process are longer. In this paper, we assume that the error terms are autocorrelated and have an autoregressive of order one, AR(1). This case is a more general case of the assumption that the error terms are identically and independently normally distributed. Since when the error terms are uncorrelated over the time, the estimate of the parameter of AR(1) is approximately zero.
If the parameter of AR(1) is unknown, one can estimate it from the data set. Using two real data sets, the model parameters are estimated and compared with the case when the error terms are independent and identically distributed. Such computations are available by using procedures AUTOREG and model in SAS. Computations show that an AR(1) can be used as a useful tool to remove the autocorrelation between the residuals.
References
Al-Haj Ebrahem, M. (2007). Estimating the variance components of accelerated degradation models. Applied Statistical Science, 15, 191-198.
Al-Haj Ebrahem, M., and Higgins, J. (2005). Non-parametric analysis of a proportional wearout model for accelerated degradation data. Applied Mathematics and Computation,
, 365-373.
Brockwell, P. J., and Davis, R. A. (2003). Introduction to Time Series and Forecasting. Springer.
Fuller, W. A. (1996). Introduction to Statistical Time Series. John Wiley and Sons.
Gallant, A. R. (1987). Nonlinear Statistical Models. New York: John Wiley and Sons.
Hudak, S. J. J., Saxena, A., Bucci, R. J., and Malcolm, R. C. (1978). Development of standard methods of testing and analyzing fatigue crack growth rate data (Tech. Rep.). Westinghouse Electric Corporation, Pittsburgh, PA 15235: AFML-TR-78-40, Westinghouse R&D Center.
Lu, J., and Meeker, W. (1993). Using degradation measures to estimate a time-to-failure distribution. Technometrics, 35, 161-174.
Lu, J., Meeker,W., and Escobar, L. (1996). A comparison of degradation and failure time analysis methods for estimating a time-to-failure distribution. Statistica Sinica, 6, 531-546.
Meeker,W., and Escobar, L. (1998). Statistical Methods for Reliability Data. JohnWiley and Sons.
Meeker, W., Escobar, L., and Lu, J. (1998). Accelerated degradation tests: Modeling and analysis. Technometrics, 40, 89-99.
Meeker, W., and Hamda, M. (1995). Statistical tools for the rapid development and evaluation of high-reliability products. IEEE Transactions on Reliability, 44, 187-198.
Oliveria, V., and Colosimo, E. (2004). Comparison of methods to estimate the timeto failure distribution in degradation tests. Quality and Reliability Engineering International, 20, 363-373.
Rawlings, J. O., Pantula, S. G., and Dickey, D. A. (2001). Applied regression analysis: A research tool (2nd ed.).
Seber, G. A. F., and Wild, C. J. (1989). Nonlinear Regression. New York: John Wiley and Sons.
Wu, S., and Shao, J. (1999). Reliability analysis using the least square method in nonlinear mixed-effect degradation models. Statistica Sinica, 9, 855-877.
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