Nonparametric, tuning-free estimation of S-shaped functions

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

LSE Research Online Documents on Economics from London School of Economics and Political Science, LSE Library

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 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.

Keywords: sequential algorithm; shape-constrained regression; s-shaped functions; S-shaped functions (search for similar items in EconPapers)
JEL-codes: C1 (search for similar items in EconPapers)
Pages: 29 pages
Date: 2022-09-01
New Economics Papers: this item is included in nep-ecm
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Citations: View citations in EconPapers (1)

Published in Journal of the Royal Statistical Society. Series B: Statistical Methodology, 1, September, 2022, 84(4), pp. 1324 - 1352. ISSN: 1369-7412

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Persistent link: https://EconPapers.repec.org/RePEc:ehl:lserod:111889

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