Nonparametric, tuningfree estimation of Sshaped functions
Abstract
We consider the nonparametric estimation of an Sshaped regression function. The least squares estimator provides a very natural, tuningfree approach, but results in a nonconvex 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 primaldual 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 worstcase 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 almostparametric 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 finitesample properties of the estimator, and our algorithm is implemented in the R package Sshaped.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.07257
 Bibcode:
 2021arXiv210707257F
 Keywords:

 Statistics  Methodology;
 Mathematics  Statistics Theory
 EPrint:
 79 pages, 10 figures