Nonparametric M-quantile Regression via Penalized Splines

Quantile regression investigates the conditional quantile functions of a response variables in terms of a set of covariates. Mquantile regression extends this idea by a “quantile-like” generalization of regression based on influence functions. In this work we extend it to nonparametric regression, in the sense that the M-quantile regression functions do not have to be assumed to be linear, but can be left undefined and estimated from the data. Penalized splines are employed to estimate them. This choice makes it easy to move to bivariate smoothing and additive modeling. An algorithm based on penalized iteratively reweighted least squares to actually fit the model is also proposed. Simulation studies are presented that show the finite sample properties of the proposed estimation technique.