Zhiqiang Liao, Sheng Dai, Eunji Lim, Timo Kuosmanen
arXiv 15 Apr 2024 · Statistics — Methodology · 1 citations (OpenAlex)
arXiv:2404.09528 · PDF · DOI · OpenAlex · Extracted main text
Convex regression is a method for estimating the convex function from a data set. This method has played an important role in operations research, economics, machine learning, and many other areas. However, it has been empirically observed that convex regression produces inconsistent estimates of convex functions and extremely large subgradients near the boundary as the sample size increases. In this paper, we provide theoretical evidence of this overfitting behavior. To eliminate this behavior, we propose two new estimators by placing a bound on the subgradients of the convex function. We further show that our proposed estimators can reduce overfitting by proving that they converge to the underlying true convex function and that their subgradients converge to the gradient of the underlying function, both uniformly over the domain with probability one as the sample size is increasing to infinity. An application to Finnish electricity distribution firms confirms the superior performance of the proposed methods in predictive power over the existing methods.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Mazumder, Choudhury, Iyengar \ Sen (2019) `A computational framework for multivariate convex regression and its variants', Journal of the American Statistical Association… | 1.000 | 7 | 3 | 100% |
| 2 | Kuosmanen (2008) `Representation theorem for convex nonparametric least squares', Econometrics Journal 11, 308–325 self | 0.843 | 3 | 3 | 100% |
| 3 | Kurdila \ Zabarankin (2006) Convex functional analysis, Springer Science & Business Media, Switzerland | 0.843 | 3 | 3 | 100% |
| 4 | Kuosmanen (2012) `Stochastic semi-nonparametric frontier estimation of electricity distribution networks: Application of the stoned method in the… self | 0.811 | 4 | 2 | 100% |
| 5 | Kuosmanen \ Johnson (2020) `Conditional yardstick competition in energy regulation', The Energy Journal 41, 67–92 | 0.811 | 4 | 2 | 100% |
| 6 | Lim \ Glynn (2012) `Consistency of multidimensional convex regression', Operations Research 60, 196–208 | 0.811 | 4 | 2 | 100% |
| 7 | Seijo \ Sen (2011) `Nonparametric least squares estimation of a multivariate convex regression function', The Annals of Statistics 39, 1633–1657 | 0.794 | 6 | 4 | 50% |
| 8 | Bertsimas \ Mundru (2021) `Sparse convex regression', INFORMS Journal on Computing 33, 262–279 | 0.737 | 3 | 2 | 100% |
| 9 | Ghosal \ Sen (2017) `On univariate convex regression', Sankhya A 79, 215–253 | 0.737 | 3 | 2 | 100% |
| 10 | Kuosmanen, Kuosmanen \ Dai (2022) Kohtuullinen muuttuva kustannus sähkön jakeluverkkoyhtiöiden valvontamallissa: Ehdotus tehostamiskannustimen kehittämiseksi 6 | 0.737 | 3 | 2 | 100% |
Showing the top 10 of 30 scored citations.
arXiv econ.EM papers that cite this one, ranked by how heavily they lean on it.
| Citing paper | Intensity | Mentions | Sections | |
|---|---|---|---|---|
| 1 | Orthogonality conditions for convex regression | 0.405 | 1 | 1 |