Inference for a Simple Step-Stress Model with Type-I Censoring and Lognormally Distributed Lifetimes

Abstract

Accelerated life-testing (ALT) is a very useful technique for examining the reliability of highly reliable products. It allows the experimenter to obtain failure data more quickly at increased stress levels than under normal operating conditions. A step-stress model is one special class of ALT, and in this article we consider a simple step-stress model under the cumulative exposure model with lognormally distributed lifetimes in the presence of Type-I censoring. We then discuss inferential methods for the unknown parameters of the model by the maximum likelihood estimation method. Some numerical methods, such as the Newton–Raphson and quasi-Newton methods, are discussed for solving the corresponding non-linear likelihood equations. Next, we discuss the construction of confidence intervals for the unknown parameters based on (i) the asymptotic normality of the maximum likelihood estimators (MLEs), and (ii) parametric bootstrap resampling technique. A Monte Carlo simulation study is carried out to examine the performance of these methods of inference. Finally, a numerical example is presented in order to illustrate all the methods of inference developed here.

Keywords:

• Accelerated life-testing
• Bootstrap method
• Coverage probabilities
• Cumulative exposure model
• Fisher information matrix
• Lognormal distribution
• Maximum likelihood estimation
• Step-stress experiment
• Type-I censoring

Mathematics Subject Classification:

• 62H10
• 62G30
• 46N30
• 65C60

Acknowledgments

The authors thank the Natural Sciences and Engineering Research Council of Canada for supporting this research and an anonymous referee for making some suggestions which led to a considerable improvement in the presentation and clarity of this manuscript. The first author also thanks the guest editors of this special issue for extending an invitation to prepare this article.