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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. 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).
    4. 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.
    5. 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.
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    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. Zhiqiang Liao, 2024. "Variable selection in convex nonparametric least squares via structured Lasso: An application to the Swedish electricity distribution networks," Papers 2409.01911, arXiv.org, revised Nov 2024.
    13. 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.
    14. 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.
    15. 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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