Nonparametric, tuning-free estimation of S-shaped functions

• 成果类型: Article
• 署名作者: Feng, Oliver Y.; Chen, Yining; Han, Qiyang; Carroll, Raymond J.; Samworth, Richard J.
• 署名单位: University of Cambridge; University of London; London School Economics & Political Science; Rutgers University System; Rutgers University New Brunswick; Texas A&M University System; Texas A&M University College Station; University of Technology Sydney
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1111/rssb.12481
• 发表日期: 2022
• 页码: 1324-1352
• 关键词: atmospheric atomic mercury; least-squares estimators; dual bases algorithm; risk bounds; unimodal regression; density; adaptation; constraints; models

摘要

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.