臺灣博碩士論文加值系統

The performance of reliability inference strongly depends on the modeling of a product’s lifetime distribution. Therefore, the effects of model mis-specification on the product’s lifetime prediction is an interesting research topic. For highly reliable products, this study addresses the effects of model mis-specification in an ALT experiment when the GG3 distribution is either mis-specified as Lognormal or Weibull distribution. We first derive the analytical expressions for the expected log likelihood function when GG3 distribution is either mis-specified as Lognormal or Weibull distribution. Then, the best parameters for the wrong model can be obtained directly via a numerical optimization. Furthermore, we also define the relative bias (RB) and relative variability (RV) to measure the accuracy and precision of the estimated p-th quantile of the product’s lifetime distribution. Both complete and censored ALT models are studied. The results demonstrate that the tail quantiles are significantly overestimated (underestimated) when data wrongly fitted by Lognormal (Weibull) distribution; while the variability of the tail quantiles significantly enlarged when data wrongly fitted by Lognormal (Weibull) distribution. Furthermore, when the sample size and censoring ratio are not large enough, a simulation study shows that the effect of model mis-specification on the tail quantiles is not negligible.

1. Introduction 3
2. A Motivating Example and Problem Formulation 5
2.1. Motivation of this study 5
2.2. Assumptions and Problem Formulation 8
3. The Effects of Model Mis-specification 11
3.1. Asymptotic Distribution of Quasi Maximum Likelihood Estimators 11
3.2. Relative Bias (RB) and Relative Variation (RV) 12
3.3. RB and RV of p-th Quantile under ALT Experiment 12
3.4. An Illustrative Example 16
3.4.1 The Effects of parameter k on the Bias and MSE 19
4. Simulation Study when the Sample Sizes are Finite 21
4.1. The Case of Complete Data 21
4.2. The Case of Censoring Data 24
5. Conclusion and Extension 27
6. Appendix. 28
Appendix 1. The proofs of (3.15) and (3.17) 28
Appendix 2. Asymptotic covariance matrix of mis-treated distributions 29
Appendix 3. The proofs of (3.25) and (3.26) 32
7. References 33

1. J. F. Lawless (1980). Inference in the generalized gamma and log gamma distribution. Technometrics, 22:409–419.
2. J. F. Lawless (1982). Statistical Models and Methods for Lifetime Data. Wiley: New York.
3. J. Lieblein and M. Zelen (1956). Statistical investigation of the fatigue life of deep groove Ball Bearings, Journal of Research of the National Bureau of Standards, 57:273-316.
4. W. Q. Meeker (1984). A comparison of accelerated life test plans for Weibull and Lognormal distributions and Type I censoring, Technometrics, 26:157–172.
5. W. Q. Meeker and L. A. Escobar (1998). Statistical Methods for Reliability Data. Wiley, New York.
6. W. Nelson (1990). Accelerated Testing: Statistical Models, Test Plans, and Data Analyses. Wiley, New York.
7. W. Nelson and W. Q. Meeker (1978). Theory for optimum censored accelerated life tests for Weibull and extreme value distributions, Technometrics, 20:171–177.
8. F. G. Pascual (2005). Maximum likelihood estimation under mis-specified Lognormal and Weibull distribution, Communications in Statistics-Simulation and Computation, 34:503–524.
9. F. G. Pascual (2006). Accelerated life test plans robust to mis-specification of the stress-life relationship, Technometrics, 48:11-25.
10. F. G. Pascual and G. Montepiedra (2005). Lognormal and Weibull Accelerated life test plans under distribution mis-specification, IEEE Transactions on Reliability, 54:43–52.
11. C. Y. Peng and S. T. Tseng (2009). Mis-specification analysis of linear degradation models, IEEE Transactions on Reliability, 58:444-454.
12. Pham and Almhana (1995). The generalized gamma distribution: its hazard rate and stress-strength model. IEEE Transactions on Reliability, 44:392-397.
13. R. L. Prentice (1974). A log gamma model and its maximum likelihood estimation, Biometrika, 61:539-544.
14. E. W. Stacy (1962). A generalization of the gamma distribution, Annals of Mathematical Statistics, 33:1187–1192.
15. C. C. Tsai, S. T. Tseng and N. Balakrishnan (2011). Mis-specification analyses of Gamma and Wiener degradation processes, Journal of Statistical Planning and Inference, 141:3725–3735.
16. H. White (1982). Maximum likelihood estimation of mis-specified models. Econometrica, 50:1–25.