Inference for cumulative risk model under step-stress experiments and its application in nanocrystalline data

With the popularity of step-stress accelerated life testing, researchers are exploring more possibilities for models that relate the life distributions under different stress levels. Cumulative risk model assumes that the effects of stress changes have a lag period before they are fully observed, which guarantees the continuity of the hazard rate function. This paper studies the cumulative risk model for Lomax distribution with step-stress experiments. For maximum likelihood estimation, Newton-Rapson method is adopted to get point estimates. Meanwhile, the asymptotic normality of the maximum likelihood estimator is used to obtain asymptotic confidence intervals. For Bayesian estimation, point estimates and highest posterior density credible intervals under squared error loss function with informative prior and non-informative prior are derived using Metropolis-Hastings method and Metropolis-Hastings within Gibbs algorithm. To evaluate the effects of stress change time and the length of lag period, as well as the performance of different methods, numerical simulations are conducted. Then a real nanocrystalline data set is analyzed.