Homeomorphic smoothing splines: a new tool for monotonizing an unconstrained estimator in nonparametric regression

In this paper we focus on nonparametric estimation of a monotone regression function using general spline smoothing techniques. We propose to construct a new class of monotone splines by using some tools that have been recently developed in the context of image warping to compute smooth and bijective deformations. Our idea is to adapt these tools to construct a monotone spline that can be used to monotonize any unconstrained estimator of a regression function. We show that under mild conditions, this monotone smoother inherits the convergence properties of the unconstrained estimator. Our estimator is defined as the solution of an ordinary differential equation governed by an appropriate time-dependent vector field depending on the unconstrained estimate. Unlike some classical spline smoothing techniques under shape constraints, our method does not require the use of quadratic programming techniques under linear constraints and has therefore a low computational cost. We illustrate its finite sample performance via a simulation study, and we also compare it with a recently proposed constrained estimator. An illustration to some real data is given.