Chapter Eight Interval-censored reliability tests under lognormal lifetimes

Dealing with censored data is an important concern in reliability and survival analysis. Interval censored data usually arises in reliability tests when failure times are known only to lie within an interval instead, but cannot be observed exactly. Moreover, modern devices often exhibit high reliability, necessitating prolonged testing periods for accurate analysis under normal operating conditions, and thus a high experimental cost. To reduce experimental times, accelerated life testing (ALT) is used as an alternative approach, effectively shortening the experimental duration by inducing failures more rapidly through the application of increased stress factors. Subsequent inferences drawn from these tests can then be extended to normal usage conditions. Among the different types of ALTs, step-stress design progressively increases the stress level at some predetermined times during the experiment, for every surviving unit under test. Step-stress has proven to be an attractive accelerated design, enabling accurate inference with low experimental units. Parametric inference usually assumes a specific parametric family of distributions modeling the lifetime of the devices. Within this context, the lognormal distribution is often adopted due to its practical relevance. Classical estimation methods rely on the likelihood function associated with the lognormal distribution to estimate its parameters, and thus the lifetime distribution function. Yet, maximum likelihood techniques are known to be significantly influenced by anomalies in the data. In this work, we introduce a new family of robust estimators based on divergence measures for the analysis of step-stress experiments subjected to interval censoring, under the assumption of lognormal lifetime distributions. We derive the asymptotic distribution of the proposed minimum density power divergence estimators (MDDPEs) and establish asymptotic confidence intervals for the model parameters and some lifetime characteristics of interest. A real-world application is also presented to demonstrate the utility and effectiveness of the model and robust inferential methods proposed.