Exact inference for progressively Type-I censored step-stress accelerated life test under interval monitoring

Thanks to continuously advancing technology and manufacturing processes, the products and devices are becoming highly reliable but performing the life tests of these products at normal operating conditions has become extremely difficult, if not impossible, due to their long lifespans. This problem is solved by accelerated life tests where the test units are subjected to higher stress levels than the normal usage level so that information on the lifetime parameters can be obtained more quickly. The lifetime at the design condition is then estimated through extrapolation using a regression model. Although continuous inspection of the exact failure times is an ideal mode, the exact failure times of test units may not be available in practice due to technical limitations and/or budgetary constraints, but only the failure counts are collected at certain time points during the test (i.e., interval inspection). In this work, we consider the progressively Type-I censored step-stress accelerated life test under the assumption that the lifetime of each test unit is exponentially distributed. Under this setup, we obtain the maximum likelihood estimator of the mean time to failure at each stress level and derive its exact sampling distribution under the condition that its existence is ensured. Using the exact distribution of the MLE as well as its asymptotic distribution and the parametric bootstrap method, we then discuss the construction of confidence intervals for the mean parameters and their performance is assessed through Monte Carlo simulations. Finally, an example is presented to illustrate all the methods of inference discussed here.