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Consistency of Multidimensional Convex Regression

Author

Listed:
  • Eunji Lim

    (Department of Industrial Engineering, University of Miami, Coral Gables, Florida 33124)

  • Peter W. Glynn

    (Department of Management Science and Engineering, Stanford University, Stanford, California 94305)

Abstract

Convex regression is concerned with computing the best fit of a convex function to a data set of n observations in which the independent variable is (possibly) multidimensional. Such regression problems arise in operations research, economics, and other disciplines in which imposing a convexity constraint on the regression function is natural. This paper studies a least-squares estimator that is computable as the solution of a quadratic program and establishes that it converges almost surely to the “true” function as n (rightarrow) (infinity) under modest technical assumptions. In addition to this multidimensional consistency result, we identify the behavior of the estimator when the model is misspecified (so that the “true” function is nonconvex), and we extend the consistency result to settings in which the function must be both convex and nondecreasing (as is needed for consumer preference utility functions).

Suggested Citation

  • Eunji Lim & Peter W. Glynn, 2012. "Consistency of Multidimensional Convex Regression," Operations Research, INFORMS, vol. 60(1), pages 196-208, February.
    • Handle: RePEc:inm:oropre:v:60:y:2012:i:1:p:196-208
    DOI: 10.1287/opre.1110.1007
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    References listed on IDEAS

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    1. Adonis Yatchew & Len Bos, 1997. "Nonparametric Least Squares Regression and Testing in Economic Models," Working Papers yatchew-99-01, University of Toronto, Department of Economics.
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    Citations

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    Cited by:

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    3. Wang, Yongqiao & Wang, Shouyang & Dang, Chuangyin & Ge, Wenxiu, 2014. "Nonparametric quantile frontier estimation under shape restriction," European Journal of Operational Research, Elsevier, vol. 232(3), pages 671-678.
    4. Nanjing Jian & Shane G. Henderson, 2020. "Estimating the Probability that a Function Observed with Noise Is Convex," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 376-389, April.
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    6. 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.
    7. Peter W. Glynn & Yijie Peng & Michael C. Fu & Jian-Qiang Hu, 2021. "Computing Sensitivities for Distortion Risk Measures," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1520-1532, October.
    8. Zhiqiang Liao & Zhaonan Qu, 2024. "Structured Lasso for convex nonparametric least squares: An application to Swedish electricity distribution networks," Papers 2409.01911, arXiv.org, revised Oct 2025.
    9. Kuosmanen, Timo & Johnson, Andrew, 2017. "Modeling joint production of multiple outputs in StoNED: Directional distance function approach," European Journal of Operational Research, Elsevier, vol. 262(2), pages 792-801.
    10. David J. Eckman & Matthew Plumlee & Barry L. Nelson, 2022. "Plausible Screening Using Functional Properties for Simulations with Large Solution Spaces," Operations Research, INFORMS, vol. 70(6), pages 3473-3489, November.
    11. Eunji Lim, 2014. "On Convergence Rates of Convex Regression in Multiple Dimensions," INFORMS Journal on Computing, INFORMS, vol. 26(3), pages 616-628, August.
    12. Mihaela Curmei & Georgina Hall, 2025. "Shape-Constrained Regression Using Sum of Squares Polynomials," Operations Research, INFORMS, vol. 73(1), pages 543-559, January.
    13. Ghosal, Rahul & Ghosh, Sujit & Urbanek, Jacek & Schrack, Jennifer A. & Zipunnikov, Vadim, 2023. "Shape-constrained estimation in functional regression with Bernstein polynomials," Computational Statistics & Data Analysis, Elsevier, vol. 178(C).
    14. Ryan Cory-Wright & Cristina Cornelio & Sanjeeb Dash & Bachir El Khadir & Lior Horesh, 2024. "Evolving scientific discovery by unifying data and background knowledge with AI Hilbert," Nature Communications, Nature, vol. 15(1), pages 1-14, December.
    15. Eunji Lim, 2025. "Estimating a Function and Its Derivatives Under a Smoothness Condition," Mathematics of Operations Research, INFORMS, vol. 50(2), pages 1112-1138, May.
    16. Bodhisattva Sen & Mary Meyer, 2017. "Testing against a linear regression model using ideas from shape-restricted estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(2), pages 423-448, March.
    17. Xie, Jianhui & Pan, Zhewen, 2025. "Structure identification for partially linear partially concave models," European Journal of Operational Research, Elsevier, vol. 324(1), pages 142-154.
    18. Lee, Chia-Yen & Johnson, Andrew L. & Moreno-Centeno, Erick & Kuosmanen, Timo, 2013. "A more efficient algorithm for Convex Nonparametric Least Squares," European Journal of Operational Research, Elsevier, vol. 227(2), pages 391-400.
    19. Julia Schaefer & Marcel Clermont, 2018. "Stochastic non-smooth envelopment of data for multi-dimensional output," Journal of Productivity Analysis, Springer, vol. 50(3), pages 139-154, December.
    20. Chen, Yining & S. Torrent, Hudson & A. Ziegelmann, Flavio, 2023. "Robust nonparametric frontier estimation in two steps," LSE Research Online Documents on Economics 119389, London School of Economics and Political Science, LSE Library.
    21. Henry Lam & Clementine Mottet, 2017. "Tail Analysis Without Parametric Models: A Worst-Case Perspective," Operations Research, INFORMS, vol. 65(6), pages 1696-1711, December.

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