Nonparametric M-regression with free knot splines

Many problems of practical interest can be formulated as the nonparametric estimation of a certain function such as a regression function, logistic or other generalized regression function, density function, conditional density function, hazard function, or conditional hazard function. Extended linear modeling provides a convenient theoretical framework for using polynomial splines and their selected tensor products in such function estimation problems and especially for obtaining rates of convergence of the resulting estimates in a unified manner. For a long time the theoretical results were restricted to fixed knot splines and to log-likelihood functions that were twice continuously differentiable. Recently, Stone and Huang extended the theory to handle free knot splines. In the present paper, the theory is further extended to handle contexts in which the log-likelihood function may not be differentiable. Specifically, we establish rates of convergence for estimation based on free knot splines in the context of nonparametric regression corresponding to M-estimates, which includes least absolute deviations (LAD) regression, quantile regression, and robust regression as special cases.