Estimation of a regression function with a sharp change point using boundary wavelets

We propose a sharp change point estimator based on the differences between right and left boundary wavelet smoothers. It is constructed by applying a two-step procedure to the observed data and has the minimax convergence rate. Next, we estimate the regression function with boundary wavelets in the left and right regions of the estimated jump point separately. This method helps us to capture the feature of a discontinuity in practice. Both mean integrated squared error and mean squared error of the estimated function are derived and we then show that these rates of convergence are the same as the case in which a jump point does not exist. Simulated examples demonstrate the improved performance of the proposed methods.