# Step-stress accelerated life testing with two stress factors

- Beschleunigte step-stress-Lebensdauertestung unter Berücksichtigung von zwei Stressfaktoren

Pitzen, Simon Maria; Kateri, Maria (Thesis advisor); Kamps, Udo (Thesis advisor)

*Aachen : RWTH Aachen University (2021)*

Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

In this thesis, we discussed (simple) SSALT with two stress factors for exponentially distributed lifetimes. Under the common CE-assumption and stating a log-linear life-stress relationship, we considered the role of step-stress testing in ALT in general and different specific aspects of bivariate (simple) SSALT. We proved that for a wide range of test designs for two stress factors applying arbitrary time varying or constant stress functions a statistically equivalent bivariate simple step-stress test can be constructed, resulting in the same asymptotic variance matrix as the initial test design. Therefore, if the objective of the design procedure is to optimize a criterion solely based on the Fisher information matrix as it is common in practice and research, the consideration of simple step-stress plans is sufficient. In other words, the motivation of alternative stress loadings like k-level constant stress, ramp-stress, or multiple step-stress designs requires additional alternative objectives that do not depend on the Fisher information matrix like the minimization of the probability of non-existent estimates or achieving desirable properties for small sample sizes. For given stress levels of a bivariate simple step-stress test, we were able to find the optimal change points minimizing the asymptotic variance of the logarithm of the MLE of the mean time to failure under NOC for step-up and step-down tests. The fact that the optimal change points were derived in closed form enables the derivation of a lower bound for the asymptotic variance of bivariate simple step-stress tests in general, only depending on the larger amount of extrapolation between the two stress factors. Further results on optimal test designs were deduced also considering the minimization of the non-existence probability as a secondary criterion when the asymptotic variance is fixed to a given level. In order to account for the influence of misspecified necessary pre-estimates of the model parameters on the estimation based on a resulting non-optimal test plan, the change points were derived under these circumstances. We found a representation of the asymptotic variance of the MLE under a non-optimal test design as a function of the relative amounts of misspecification. This allows to calculate the caused deviation from the minimal possible asymptotic variance based on the choice of stress levels and ranges of misspecification alone without any knowledge of the model parameters. As another important aspect in the conduction of SSALT experiments, we compared different methods to estimate the parameters of the link function between the lifetime distribution and the stress factors for type-II censored data. We proposed a least squares approach based on the MLEs of the scale parameters on the respective increased stress levels which always leads to closed form estimates and outlined prerequisites for equivalence to the standard direct ML estimation of the log-link parameters. In a situation where the two methods do not coincide, we proved the existence of the MLEs of the link function parameters and found conditions that ensure receiving the estimators in closed form as well. Furthermore, alternative consistent closed form estimators of the log-link parameters based on the MLEs of the scale parameters were motivated. The extension of known results on the distribution of these MLEs to multiple step-stress situations allowed to deduce exact distributional properties of the MLE of the mean lifetime under NOC.

### Identifier

- DOI: 10.18154/RWTH-2021-06801
- RWTH PUBLICATIONS: RWTH-2021-06801