Nonparametric, tuning-free estimation of S-shaped functions

Oliver Feng, Yining Chen, Qiyang Han, Raymond J. Carroll, Richard J. Samworth

Research output: Contribution to journalArticlepeer-review

1 Citation (SciVal)


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 optimization 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.
Original languageEnglish
Pages (from-to)1324-1352
JournalJournal of the Royal Statistical Society, Series B (Statistical Methodology)
Issue number4
Early online date21 Apr 2022
Publication statusPublished - 21 Apr 2022


Dive into the research topics of 'Nonparametric, tuning-free estimation of S-shaped functions'. Together they form a unique fingerprint.

Cite this