Nonparametric, tuning-free estimation of S-shaped functions

TL;DR

This paper introduces a tuning-free, nonparametric estimator for S-shaped functions that is computationally efficient, theoretically optimal, and adaptable to various function complexities, with practical implementation in R.

Contribution

It proposes a novel projection-based estimator for S-shaped functions that is both computationally feasible and theoretically optimal, including an efficient algorithm and an R package.

Findings

Achieves minimax optimal convergence rates for function and inflection point estimation.

Demonstrates robustness to model misspecification.

Shows strong finite-sample performance through simulations and real data.

Abstract

We consider the nonparametric estimation of an S-shaped regression function. The least squares estimator provides a very natural, tuning-free approach, but results in a non-convex optimisation problem, since the inflection point is unknown. We show that the estimator may nevertheless be regarded as a projection onto a finite union of convex cones, which allows us to propose a mixed primal-dual bases algorithm for its efficient, sequential computation. After developing a projection framework that demonstrates the consistency and robustness to misspecification of the estimator, our main theoretical results provide sharp oracle inequalities that yield worst-case and adaptive risk bounds for the estimation of the regression function, as well as a rate of convergence for the estimation of the inflection point. These results reveal not only that the estimator achieves the minimax optimal rate of convergence for both the estimation of the regression function and its inflection point (up to a logarithmic factor in the latter case), but also that it is able to achieve an almost-parametric rate when the true regression function is piecewise affine with not too many affine pieces. Simulations and a real data application to air pollution modelling also confirm the desirable finite-sample properties of the estimator, and our algorithm is implemented in the R package Sshaped.