Rate of Convergence of a Change Point Estimator in a Misspecified Regression Model

A parametric estimation problem is considered in a misspecified regression model, where the regression function has a smooth change point. The focus lies on regression functions, which are continuous at the change point. Here, it is not assumed that the true regression function belongs to the model class. However, there exists a pseudo change point, such that the related regression function gives a reasonable approximation. With increasing sample size the asymptotic behavior is investigated of the least squares estimates of the change point. The consistency of the change point estimator for the pseudo estimator is shown. It turns out that the rate of convergence depends on the order of smoothness of the regression function at the change point.