Contributions to Modeling and Inference for k-Level Step Stress Accelerated Life Tests under Progressive Type-I Censoring with Lifetimes for a Log-Location Scale Family

In reliability engineering, the accelerated life test is not only becoming increasingly popular but also absolutely necessary. This is due to the rapid yield of information about the lifetime distribution of highly reliable products and devices in shorter time periods, which it achieves by conducting the life test at more extreme stress levels than normal operating conditions. Through extrapolation, the lifetime distribution at the usage stress can be estimated with an appropriate regression model. In our work, we inspect a number of modeling and inferential problems related to censored lifetime data from log-location-scale distribution under the step-stress accelerated life tests (ssALT). First, we investigate the inference of a progressively Type-I censored k-level step-stress accelerated life test when the lifetime of a test unit follows a log-location-scale family of distributions. Although simple and analytical, the popular exponential distribution lacks the model flexibility desired in practice due to the constraint of constant hazard rates. Pragmatically, Weibull or lognormal distributions, which are members of the log-location-scale family, demonstrate superior model fits. Therefore, our study is extended to consider the general log-location-scale family, and our inferential, prescriptive methods are illustrated using three popular lifetime distributions, including Weibull, lognormal, and log-logistic. The log-location-scale family provides stronger model flexibility, hence provides better fits and improves the reliability analysis considerably. Assuming that the location parameter is linearly linked to the (transformed) stress level (i.e., μi = α + βxi), an iterative algorithm is developed to estimate the regression parameters α and β along with the scale parameter σ. Subsequently, this allows the intermediate censoring to take place at the end of each stress level xi (viz.,τi, i = 1,...,k). Furthermore, using the exact distributions of the estimators, the Wald-type asymptotic 95% confidence interval and BCa bootstrap 95% confidence intervals for the respective parameters are obtained for comparison. Then, we determine the optimal stress duration times numerically under various design criteria based on Fisher's expected information, namely D-optimality, T-optimality, C-optimality, A-optimality, M-optimality, and E-optimality. The effect of intermediate censoring proportion on design efficiency is also assessed computationally with a real engineering case study into analyzing the reliability characteristic of a solar lighting device. Next, we study the statistical inference for progressively Type-I censored k-level step-stress accelerated life tests under interval monitoring, with the lifetime of a test unit following a log-location scale family distributions, and compared simulation studies with previous continuous monitoring results. Finally, we formulate the cost model function for the constrained optimal designs under simple ssALT and then define five design criteria based on Fisher's expected information and study the efficiency of optimality models under constrained and unconstrained settings.