Analysis of change-point estimators under the null hypothesis

Abstract

We consider estimators for the change-point in a sequence of independent observations. These are defined as the maximizing points of weighted U-statistic type processes. Our investigations focus on the behaviour of the estimators in the case of independent and identically distributed random variables (null hypothesis of no change), but contiguous alternatives in the sense of Oosterhoff and van Zwet are also taken into account. If the weight functions belong to the Chibisov-O'Reilly class we derive convergence in distribution, including a special Berry-Esseen result. The limit variable is the almost sure unique maximizing point of a weighted (standard or reflected) Brownian bridge with drift. For general weight functions the limiting null distribution is analytically not known. However, in the special case where no weight functions are involved it is known that the maximizer of a standard Brownian bridge is uniformly distributed on the unit interval. A corresponding result for the reflected Brownian bridge seems to be unknown in the literature. In this paper we fill this gap and actually compute the common density of the maximum and its location for a reflected Brownian bridge. From this one can find the density of the maximizer, which analytically can be expressed in terms of a series. In a special case even the finite sample size distribution of our estimator is established. Besides distributional results, we also determine the almost sure set of cluster points.