Dynamic multivariate quantile inactivity time and applications in investigation of a treatment effect

To investigate potentially dependent lifetimes, it is necessary to extend the $ \alpha $-quantile inactivity time to bivariate and multivariate frameworks. To extend this measure to a dynamic multivariate framework, all possible trajectories at time $ t $ are considered. The behavior of the extended $ \alpha $-quantile of inactivity time was investigated in relation to the corresponding multivariate hazard rate function. The $ \alpha $-quantile of the inactivity order is defined and discussed for the multivariate case. The difference between the two bivariate $ \alpha $-quantile functions of inactivity, which is an important measure for studying the effect of treatment on lifespan, was also investigated. This measure was used to analyze the effect of laser treatment on the delay of blindness. Two bootstrap approaches were implemented to construct confidence bounds for the difference measure.