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Consistency of Penalized Convex Regression

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  • Eunji Lim

Abstract

We consider the problem of estimating an unknown convex function f_* (0, 1)^d →R from data (X1, Y1), … (X_n; Y_n).A simple approach is finding a convex function that is the closest to the data points by minimizing the sum of squared errors over all convex functions. The convex regression estimator, which is computed this way, su ers from a drawback of having extremely large subgradients near the boundary of its domain. To remedy this situation, the penalized convex regression estimator, which minimizes the sum of squared errors plus the sum of squared norms of the subgradient over all convex functions, is recently proposed. In this paper, we prove that the penalized convex regression estimator and its subgradient converge with probability one to f_* and its subgradient, respectively, as n → ∞, and hence, establish the legitimacy of the penalized convex regression estimator.

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  • Eunji Lim, 2021. "Consistency of Penalized Convex Regression," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 10(1), pages 1-69, January.
  • Handle: RePEc:ibn:ijspjl:v:10:y:2021:i:1:p:69
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    References listed on IDEAS

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    1. Rahul Mazumder & Arkopal Choudhury & Garud Iyengar & Bodhisattva Sen, 2019. "A Computational Framework for Multivariate Convex Regression and Its Variants," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(525), pages 318-331, January.
    2. Varian, Hal R, 1984. "The Nonparametric Approach to Production Analysis," Econometrica, Econometric Society, vol. 52(3), pages 579-597, May.
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    • R00 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General - - - General
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