Competing Risks Step-Stress Model with Lagged Effect under Gompertz Distribution

In many survival analysis studies, the failure of a product may be attributed to one of several competing risks. In addition, if survival time is long, researchers can adopt accelerated life tests, causing devices to fail more quickly. One popular type of accelerated life tests is the step-stress test, and in this test, the stress level changes at a predetermined point time. The manner that stress levels change abruptly and increase discontinuously has been studied extensively. This paper considers a more realistic situation where the effect of stress increases cannot be achieved all at once, but with a lag time, and we propose a step-stress model consisting of two independent competing risks with a lag period in which the failure time caused by different risks at different stress levels obey Gompertz distribution, and the range of lag period is predetermined. The unknown parameters are estimated by maximum likelihood estimation and least squares estimation. For comparison, asymptotic confidence intervals and percentile bootstrap confidence intervals are constructed. By using Monte-Carlo simulations, we obtain the means and mean square errors of the maximum likelihood estimates and the least squares estimates, as well as the mean lengths and coverage rates of the two confidence intervals, which show the performance of various methods. Finally, in order to illustrate the model and proposed methods, we analyze a dataset from a solar energy experiment.