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Title: Bootstrapped and kernel-type estimators of reliability indicators in semi-Markov processes Authors:  Eirini Votsi - Le Mans University (France) [presenting]
Salim Bouzebda - Universite de Technologie de Compiegne (France)
Abstract: Semi-Markov processes are stochastic processes that are widely used in reliability and related fields. They generalize both jump Markov processes and renewal processes. We consider semi-Markov processes in continuous-time and finite state space. The stochastic behavior of such processes is governed by the semi-Markov kernel. Empirical estimators of the semi-Markov kernel and its functionals have been proposed. The limiting distributions of such estimators have usually complicated expressions, and therefore explicit computation in practice is rather infeasible. To overcome this difficulty, we propose a general bootstrap of empirical semi-Markov kernels and of the conditional transition distributions. We consider a general bootstrap that allows for a unified treatment for resampling methods and provides a flexible framework to handle practical problems. In particular, we present three different types of estimators for the semi-Markov kernel and its functionals: the bootstrapped, the kernel-type and the bootstrapped kernel-type estimators. We further establish their asymptotic properties and focus on reliability indicators, such as the functions of reliability, availability and maintainability, as well as different failure rates. The asymptotic properties of the latter indicators are obtained by means of martingale techniques. The advantages of the use of such estimators are discussed.