Rates of Convergence of Estimated Change Points with Applications in Reliability

Abstract

We consider two models in reliability which deal with a change of the characteristics of a point process, namely a burn-in and a detection problem. Given a certain valuation structure both models lead to an optimization problem which can be viewed as an optimal stopping problem in continuous time. The solutions of these optimal stopping problems are derived by means of semimartingale representations of the underlying stochastic processes. The resulting optimal stopping times involve model parameters which are in general unknown. Using estimates of these model parameters leads to so-called estimated change points. It is investigated under which conditions and at which rate consistency of parameter estimates carries over to the corresponding sequence of estimated change points.