Weibull Step-Stress Model with a Lagged Effect

Abstract

SYNOPTIC ABSTRACTIn survival analysis and reliability, researchers are often interested in assessing the effects of different stress factors on the lifetime of experimental units. The model introduced in this article is motivated by a study of the effects of altitude and other risk factors on decompression sickness, a condition encountered when individuals are exposed to significant changes in environmental pressure. Unlike standard life-testing experiments, in this study, the levels of the stress factor, viz. altitude, are changed during the exposure duration. This is known as a step-stress test, a class of accelerated testing, widely used in material testing. Recently Kannan, Kundu, and Balakrishnan (2010) introduced the cumulative risk model as an alternative to the widely used cumulative exposure model. The new model allows for the inclusion of a lag period in the hazard function, a more realistic assumption in most applications. In this article, we consider the cumulative risk model assuming that the lifetime distributions of the experimental units follow Weibull distributions at the different levels of the risk factor. It is assumed that the level of the stress factor is changed only once during the exposure duration at a pre-fixed time τ1. The maximum likelihood and the least squares methods have been used to estimate the unknown parameters. Monte Carlo simulations are performed to compare the performances of the two different methods. We further propose the Bayes estimators of the unknown parameters of the model. To evaluate the performance of the model, one data set from the altitude decompression sickness experiment has been analyzed, and it is observed that the proposed model fits the data quite well.