Robust Statistical Methods for Step-Stress Accelerated Life Test Data

Abstract

Accelerated Life Testing (ALT) is commonly implemented to derive insights into experimental items' reliability. Conducting tests under typical usage conditions can be both time-consuming and expensive. In ALT, an experimental item is examined under levels of physical stress, such as temperature, voltage, or pressure, higher than the experimental item will experience under normal operation levels. Step-stress accelerated life testing (SSALT) is a special type of accelerated life test designed to gradually increase stress levels, thereby accelerating the failure process and allowing for data collection in a shorter period.

The main contribution of this thesis lies in the development of predictive and robust methods based on imprecise probability theories. These methods provide interval-based results that reflect uncertainty in both the data and the model, unlike traditional approaches that rely on exact values and strong assumptions. By allowing for imprecision, the proposed methods offer more realistic and cautious predictions, which are necessary when normal stress data is limited or the model is uncertain.

This thesis presents three robust statistical methods for the analysis of SSALT data based on theories of imprecise probability. In the first method, imprecision is incorporated based on the likelihood ratio test within the cumulative exposure model. This method consists of three steps. First, failure times occurring under different strategies at higher stress levels are transformed to the normal stress level. Second, imprecision is introduced based on the likelihood ratio test applied to the accelerating parameter under the null hypothesis that all failure times originate from the same distribution. This imprecision allows failure times to be transformed into interval values at the normal stress level, where the transformed failure times are assumed to be indistinguishable from those observed under normal conditions. Third, Nonparametric Predictive Inference (NPI) is applied to the transformed data to provide robust predictive inference.


In the second method, imprecision is introduced based on the log-rank test applied to a parametric link function. This imprecision facilitates the transformation of data from higher stress levels into interval-valued observations at the normal stress level. The transformation is performed using the parametric link function framework, after which the transformed data are combined with Nonparametric Predictive Inference (NPI) at the normal stress level to construct lower and upper survival functions. These methods incorporate imprecision to enhance robustness with regard to model assumptions. The results demonstrate that imprecision increases for observations derived from higher stress levels, leading to more imprecise data at the normal stress level.




In the third method, a robust Bayesian framework is developed to analyze SSALT data while incorporating imprecision in prior knowledge. This method models uncertainty by considering a class of prior distributions, where the extreme bounds of this class reflect minimal prior information about the model parameters. By using these extreme bounds, lower and upper posterior predictive distributions are derived separately, enabling the prediction of future failure times at the normal stress level. This approach also facilitates the construction of lower and upper predictive survival functions, ensuring robustness in predictive inference under model uncertainty.


The performance of the proposed methods is evaluated through simulation studies. The findings indicate that imprecision increases across all methods when the assumed likelihood function or link function is misspecified. Among the three methods, the robust Bayesian approach exhibits relatively more imprecision under model misspecification, whereas the first two methods primarily show increased imprecision due to data from higher stress strategies. Additionally, the results demonstrate that as the number of observations increases, the imprecision decreases, highlighting the impact of sample size on predictive performance.