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Estimating Smooth and Convex Functions

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  • Eunji Lim
  • Kihwan Kim

Abstract

We propose a new method for estimating an unknown regression function $f_*-[\alpha, \beta] \rightarrow \mathbb{R}$ from a dataset $(X_1, Y_1), \dots, (X_n,$ $Y_n)$ when the only information available on $f_*$ is the fact that $f_*$ is convex and twice differentiable. In the proposed method, we fit a convex function to the dataset that minimizes the sum of the roughness of the fitted function and the average squared differences between the fitted function and $f_*$. We prove that the proposed estimator can be computed by solving a convex quadratic programming problem with linear constraints. Numerical results illustrate the superior performance of the proposed estimator compared to existing methods when i) $f_*$ is the price of a stock option as a function of the strike price and ii) $f_*$ is the steady-state mean waiting time of a customer in a single server queue.

Suggested Citation

  • 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.
  • Handle: RePEc:ibn:ijspjl:v:9:y:2020:i:5:p:40
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    References listed on IDEAS

    as
    1. Ait-Sahalia, Yacine & Duarte, Jefferson, 2003. "Nonparametric option pricing under shape restrictions," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 9-47.
    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. Zheng Li & Guannan Liu & Qi Li, 2017. "Nonparametric Knn estimation with monotone constraints," Econometric Reviews, Taylor & Francis Journals, vol. 36(6-9), pages 988-1006, October.
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    More about this item

    JEL classification:

    • R00 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General - - - General
    • Z0 - Other Special Topics - - General

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