Bootstrap of Reliability Indicators for Semi-Markov Processes

The present paper concerns bootstrap and kernel-type estimators of some functionals of the semi-Markov kernel (SMK) coming from the area of reliability. This type of functionals could be found in real-world applications encountered in different scientific fields. First, the weak convergence of the bootstrap kernel-type estimators of the SMK and the conditional sojourn time distributions are proved. The convergence in probability and the asymptotic normality of their derivatives are obtained. Second, new estimators of reliability indicators such as the mean time to failure, the reliability and the availability are introduced. These estimators are based on exchangeable weighted bootstrap approaches, kernel-type approaches or both of them. The method of bootstrap is employed since it is widely used to solve problems in statistics related to limiting distributions. The bootstrap estimators are shown to be asymptotically normal. The asymptotic properties of the estimators are illustrated by means of Monte-Carlo simulations.