Mis-specification Analyses of Nonlinear Wiener Process-based Degradation Models

For some highly reliable products, degradation data have been studied quite extensively to evaluate their reliability characteristics. However, the accuracy of evaluation results depends strongly on the suitability of the proposed degradation model for capturing the degradation over time. If the degradation model is mis-specified, it may result in inaccurate results. In this work, we focus on the issue of model mis-specification between nonlinear Wiener process-based degradation models in which both the product-to-product variability and the temporal uncertainty of the degradation can be considered simultaneously with the nonlinearity in degradation paths. Specifically, a generalized Wiener process-based degradation model is wrongly fitted by its two limiting cases. The effects of model mis-specification in such situations on the MTTF (mean-time-to-failure) of the product are measured with the relative bias and the relative variability. Results from a numerical example concerning fatigue cracks show that the effect of mis-specification is serious under some parameter settings, i.e., the relative bias departs from 0, and the relative variability significantly departs from 1, if the generalized Wiener degradation process is wrongly assumed to be its limiting cases.     

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.     

A unified confidence interval for reliability-related quantities of two-parameter Weibull distribution

Statistical inference methods for the Weibull parameters and their functions usually depend on extensive tables, and hence are rather inconvenient for the practical applications. In this paper, we propose a general method for constructing confidence intervals for the Weibull parameters and their functions, which eliminates the need for the extensive tables. The method is applied to obtain confidence intervals for the scale parameter, the mean-time-to-failure, the percentile function, and the reliability function. Monte-Carlo simulation shows that these intervals possess excellent finite sample properties, having coverage probabilities very close to their nominal levels, irrespective of the sample size and the degree of censorship.