A note on estimating the change-point of a gradually changing stochastic process

Abstract

We consider an estimator of the change-point of a stochastic process satisfying some weak invariance principles. Making use of the known asymptotics of the approximating Wiener processes we derive various limiting distributions according to different orders of magnitude of the underlying change. The results take into account, but also extend those of Hušková (J. Statist. Plann. Infer. 76 (1999) 109–125), who studied a location model for gradual changes with independent, identically distributed (iid) errors. Aim of this note is to show that corresponding results hold also true in our more general setting.