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On Univariate Convex Regression

Author

Listed:
  • Promit Ghosal

    (Columbia University)

  • Bodhisattva Sen

    (Columbia University)

Abstract

We find the local rate of convergence of the least squares estimator (LSE) of a one dimensional convex regression function when (a) a certain number of derivatives vanish at the point of interest, and (b) the true regression function is locally affine. In each case we derive the limiting distribution of the LSE and its derivative. The pointwise limiting distributions depend on the second and third derivatives at 0 of the “invelope function” of the integral of a two-sided Brownian motion with polynomial drifts. We also investigate the inconsistency of the LSE and the unboundedness of its derivative at the boundary of the domain of the covariate space. An estimator of the argmin of the convex regression function is proposed and its asymptotic distribution is derived. Further, we present some new results on the characterization of the convex LSE that may be of independent interest.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:sankha:v:79:y:2017:i:2:d:10.1007_s13171-017-0104-8
    DOI: 10.1007/s13171-017-0104-8
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    References listed on IDEAS

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    1. Facer, Matthew R. & Müller, Hans-Georg, 2003. "Nonparametric estimation of the location of a maximum in a response surface," Journal of Multivariate Analysis, Elsevier, vol. 87(1), pages 191-217, October.
    2. Balabdaoui, Fadoua & Rufibach, Kaspar, 2008. "A second Marshall inequality in convex estimation," Statistics & Probability Letters, Elsevier, vol. 78(2), pages 118-126, February.
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    Cited by:

    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. 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.
    3. Yu-Chang Chen & Haitian Xie, 2022. "Personalized Subsidy Rules," Papers 2202.13545, arXiv.org, revised Mar 2022.
    4. Liao, Zhiqiang & Dai, Sheng & Kuosmanen, Timo, 2024. "Convex support vector regression," European Journal of Operational Research, Elsevier, vol. 313(3), pages 858-870.

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