Statistical inference of an $\alpha$-quantile past lifetime function with applications

In reliability engineering and survival analysis, quantile functions are fundamental and often the most natural way to represent probability distributions and data samples. In this paper, the $\alpha$-quantile function of past lifetime was estimated for right-censored data by applying the Kaplan-Meier survival estimator. The weak convergence of the proposed estimator to a Gaussian process was investigated. A confidence interval for the $\alpha$-quantile of the past life function that does not depend on the density function was proposed. The strong convergence of the estimator to a Gaussian process was also discussed. The properties of the estimator and the confidence interval were investigated in a simulation study. Finally, two real datasets were analyzed.