Estimation of piecewise-smooth functions by amalgamated bridge regression splines

We consider nonparametric estimation of a one-dimensional piecewise-smooth function observed with white Gaussian noise on an interval. We propose a two-step estimation procedure, where one first detects jump points by a wavelet based procedure and then estimates the function on each smooth segment separately by bridge regression splines. We prove the asymptotic optimality (in the minimax sense) of the resulting amalgamated bridge regression spline estimator and demonstrate its efficiency on simulated and real data examples.