Nonparametric Regression Under Qualitative Smoothness Assumptions

We propose a new nonparametric regression estimate. In contrast to the traditional approach of considering regression functions whose mth derivatives lie in a ball in the L_∞ or L₂ norm, we consider the class of functions whose $(m - 1)$st derivative consists of at most k monotone pieces. For many applications this class seems more natural than the classical ones. The least squares estimator of this class is studied. It is shown that the speed of convergence is as fast as in the classical case.