Mis-specification analyses of gamma and Wiener degradation processes

Degradation models are widely used these days to assess the lifetime information of highly reliable products if there exist some quality characteristics (QC) whose degradation over time can be related to the reliability of the product. In this study, motivated by a laser data, we investigate the mis-specification effect on the prediction of product's MTTF (mean-time-to-failure) when the degradation model is wrongly fitted. More specifically, we derive an expression for the asymptotic distribution of quasi-MLE (QMLE) of the product's MTTF when the true model comes from gamma degradation process, but is wrongly assumed to be Wiener degradation process. The penalty for the model mis-specification can then be addressed sequentially. The result demonstrates that the effect on the accuracy of the product's MTTF prediction strongly depends on the ratio of critical value to the scale parameter of the gamma degradation process. The effects on the precision of the product's MTTF prediction are observed to be serious when the shape and scale parameters of the gamma degradation process are large. We then carry out a simulation study to evaluate the penalty of the model mis-specification, using which we show that the simulation results are quite close to the theoretical ones even when the sample size and termination time are not large. For the reverse mis-specification problem, i.e., when the true degradation is a Wiener process, but is wrongly assumed to be a gamma degradation process, we carry out a Monte Carlo simulation study to examine the effect of the corresponding model mis-specification. The obtained results reveal that the effect of this model mis-specification is negligible.