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On Univariate Convex Regression

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Abstract

We find the local rate of convergence of the least squares estimator (LSE) of a one dimensional convex regression function when (a) a certain number of derivatives vanish at the point of interest, and (b) the true regression function is locally affine. In each case we derive the limiting distribution of the LSE and its derivative. The pointwise limiting distributions depend on the second and third derivatives at 0 of the “invelope function” of the integral of a two-sided Brownian motion with polynomial drifts. We also investigate the inconsistency of the LSE and the unboundedness of its derivative at the boundary of the domain of the covariate space. An estimator of the argmin of the convex regression function is proposed and its asymptotic distribution is derived. Further, we present some new results on the characterization of the convex LSE that may be of independent interest.

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Authors and Affiliations

  1. Columbia University, 1255 Amsterdam Avenue, New York, NY, 10027, USA

    Promit Ghosal & Bodhisattva Sen

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  1. Promit Ghosal
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  2. Bodhisattva Sen
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Correspondence to Promit Ghosal.

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Supported by NSF grant DMS-1150435

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Ghosal, P., Sen, B. On Univariate Convex Regression. Sankhya A 79, 215–253 (2017). https://doi.org/10.1007/s13171-017-0104-8

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  • DOI: https://doi.org/10.1007/s13171-017-0104-8

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