Two-stage change-point estimators in smooth regression models

We consider a fixed design regression model where the regression function is assumed to be smooth, i.e., Lipschitz continuous, except for a point where it has only one-sided limits and a local discontinuity occurs. We propose a two-step estimator for the location of this change point and study its asymptotic convergence properties. In a first step, initial pilot estimates of the change point and associated asymptotically shrinking intervals which contain the true change point with probability converging to 1 are obtained. In the second step, a weighted mean difference depending on the assumed location of the change point is maximized within these intervals and the maximizing argument is then the final change point estimator. It is shown that this estimator attains the rate Op(n-1) in the fixed jump case. In the contiguous case, the estimator attains the rate Op(n-1 [Delta]n-2), where [Delta]n is the sequence of jump sizes which in this case is assumed to converge to 0. For the contiguous case an invariance principle is established. A sequence of appropriately scaled deviation processes is shown to converge to a two-sided Brownian motion with triangular drift.