Convergence rate and bahadur type representation of general smoothing spline M-estimates

This work was motivated by Cox and O’Sullivan (1990) who derived the optimal convergence rates for smoothing spline estimates when the loss function is sufficiently smooth. However, the study of statistical estimates resulting from nonsmooth criteria functions has become popular in recent years. In this paper, we will study the asymptotic properties of the smoothing spline estimates when the criteria functions are insufficiently smooth. Here, the smoothing spline estimate is defined as an approximate solution to an M-estimating equation. We prove that if the derivative of loss function is Lipschitz, then the convergence rate and Bahadur type representation of the estimate can be derived simultaneously. For a specific class of loss functions with discontinuous derivatives, the Bahadur type representation is also presented provided that we know the convergence rate. Examples are given when Huber’s robust loss and median loss are employed.