Nonparametric, tuning-free estimation of S-shaped functions
Publication Date
2022Journal Title
Journal of the Royal Statistical Society. Series B: Statistical Methodology
ISSN
1369-7412
Publisher
Oxford University Press (OUP)
Language
en
Type
Article
This Version
AO
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Metadata
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Feng, O., Chen, Y., Han, Q., Carroll, R., & Samworth, R. (2022). Nonparametric, tuning-free estimation of S-shaped functions. Journal of the Royal Statistical Society. Series B: Statistical Methodology https://doi.org/10.1111/rssb.12481
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.
Keywords
ORIGINAL ARTICLE, ORIGINAL ARTICLES, sequential algorithm, shape‐constrained regression, S‐shaped functions
Sponsorship
Engineering and Physical Sciences Research Council (EP/N031938/1)
Engineering and Physical Sciences Research Council (EP/P031447/1)
Identifiers
rssb12481
External DOI: https://doi.org/10.1111/rssb.12481
This record's URL: https://www.repository.cam.ac.uk/handle/1810/336360
Rights
Licence:
http://creativecommons.org/licenses/by/4.0/
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