IDEAS home Printed from https://ideas.repec.org/a/bla/jorssb/v84y2022i4p1324-1352.html
   My bibliography  Save this article

Nonparametric, tuning‐free estimation of S‐shaped functions

Author

Listed:
  • Oliver Y. Feng
  • Yining Chen
  • Qiyang Han
  • Raymond J. Carroll
  • Richard J. Samworth

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.

Suggested Citation

  • Oliver Y. Feng & Yining Chen & Qiyang Han & Raymond J. Carroll & Richard J. Samworth, 2022. "Nonparametric, tuning‐free estimation of S‐shaped functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 84(4), pages 1324-1352, September.
  • Handle: RePEc:bla:jorssb:v:84:y:2022:i:4:p:1324-1352
    DOI: 10.1111/rssb.12481
    as

    Download full text from publisher

    File URL: https://doi.org/10.1111/rssb.12481
    Download Restriction: no

    File URL: https://libkey.io/10.1111/rssb.12481?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Stout, Quentin F., 2008. "Unimodal regression via prefix isotonic regression," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 289-297, December.
    2. Daisuke Yagi & Yining Chen & Andrew L. Johnson & Timo Kuosmanen, 2020. "Shape-Constrained Kernel-Weighted Least Squares: Estimating Production Functions for Chilean Manufacturing Industries," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 38(1), pages 43-54, January.
    3. Ginsberg, William, 1974. "The multiplant firm with increasing returns to scale," Journal of Economic Theory, Elsevier, vol. 9(3), pages 283-292, November.
    4. Promit Ghosal & Bodhisattva Sen, 2017. "On Univariate Convex Regression," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 215-253, August.
    5. Yining Chen & Richard J. Samworth, 2016. "Generalized additive and index models with shape constraints," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(4), pages 729-754, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Feng, Oliver Y. & Chen, Yining & Han, Qiyang & Carroll, Raymond J & Samworth, Richard J., 2022. "Nonparametric, tuning-free estimation of S-shaped functions," LSE Research Online Documents on Economics 111889, London School of Economics and Political Science, LSE Library.
    2. Liao, Zhiqiang & Dai, Sheng & Kuosmanen, Timo, 2024. "Convex support vector regression," European Journal of Operational Research, Elsevier, vol. 313(3), pages 858-870.
    3. Catherine Bobtcheff & Christian Gollier & Richard Zeckhauser, 2008. "Resource allocations when projects have ranges of increasing returns," Journal of Risk and Uncertainty, Springer, vol. 37(1), pages 93-93, August.
    4. AgralI, Semra & Geunes, Joseph, 2009. "Solving knapsack problems with S-curve return functions," European Journal of Operational Research, Elsevier, vol. 193(2), pages 605-615, March.
    5. Cristina Polo & Julián Ramajo & Alejandro Ricci‐Risquete, 2021. "A stochastic semi‐non‐parametric analysis of regional efficiency in the European Union," Regional Science Policy & Practice, Wiley Blackwell, vol. 13(1), pages 7-24, February.
    6. Cui, Zhenyu & Lee, Chihoon & Zhu, Lingjiong & Zhu, Yunfan, 2021. "Non-convex isotonic regression via the Myersonian approach," Statistics & Probability Letters, Elsevier, vol. 179(C).
    7. Westerink-Duijzer, L.E. & Schlicher, L.P.J. & Musegaas, M., 2019. "Fair allocations for cooperation problems in vaccination," Econometric Institute Research Papers EI2019-06, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    8. Yu-Chang Chen & Haitian Xie, 2022. "Personalized Subsidy Rules," Papers 2202.13545, arXiv.org, revised Mar 2022.
    9. Zhiqiang Liao, 2024. "Variable selection in convex nonparametric least squares via structured Lasso: An application to the Swedish electricity market," Papers 2409.01911, arXiv.org.
    10. Lee, Chia-Yen & Wang, Ke, 2019. "Nash marginal abatement cost estimation of air pollutant emissions using the stochastic semi-nonparametric frontier," European Journal of Operational Research, Elsevier, vol. 273(1), pages 390-400.
    11. Srivastava, Vaibhav & Bullo, Francesco, 2014. "Knapsack problems with sigmoid utilities: Approximation algorithms via hybrid optimization," European Journal of Operational Research, Elsevier, vol. 236(2), pages 488-498.
    12. Olesen, Ole B. & Ruggiero, John, 2012. "Maintaining the Regular Ultra Passum Law in data envelopment analysis," Discussion Papers on Economics 2/2012, University of Southern Denmark, Department of Economics.
    13. Lotty E. Westerink‐Duijzer & Loe P. J. Schlicher & Marieke Musegaas, 2020. "Core Allocations for Cooperation Problems in Vaccination," Production and Operations Management, Production and Operations Management Society, vol. 29(7), pages 1720-1737, July.
    14. Layer, Kevin & Johnson, Andrew L. & Sickles, Robin C. & Ferrier, Gary D., 2020. "Direction selection in stochastic directional distance functions," European Journal of Operational Research, Elsevier, vol. 280(1), pages 351-364.
    15. Eunji Lim & Kihwan Kim, 2020. "Estimating Smooth and Convex Functions," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 9(5), pages 1-40, September.
    16. Olesen, Ole B. & Ruggiero, John, 2014. "Maintaining the Regular Ultra Passum Law in data envelopment analysis," European Journal of Operational Research, Elsevier, vol. 235(3), pages 798-809.
    17. Aubin-Frankowski, Pierre-Cyril & Szabo, Zoltan, 2022. "Handling hard affine SDP shape constraints in RKHSs," LSE Research Online Documents on Economics 115724, London School of Economics and Political Science, LSE Library.
    18. España, Victor J. & Aparicio, Juan & Barber, Xavier & Esteve, Miriam, 2024. "Estimating production functions through additive models based on regression splines," European Journal of Operational Research, Elsevier, vol. 312(2), pages 684-699.
    19. Jose Manuel Cordero & Cristina Polo & Javier Salinas-Jiménez, 2021. "Subjective Well-Being and Heterogeneous Contexts: A Cross-National Study Using Semi-Nonparametric Frontier Methods," Journal of Happiness Studies, Springer, vol. 22(2), pages 867-886, February.
    20. Fadoua Balabdaoui & Cécile Durot & Hanna Jankowski, 2023. "Behaviour of the Monotone Single Index Model Under Repeated Measurements," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 324-350, February.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bla:jorssb:v:84:y:2022:i:4:p:1324-1352. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Wiley Content Delivery (email available below). General contact details of provider: https://edirc.repec.org/data/rssssea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.