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arXiv · 1411.4626

On Convex Least Squares Estimation when the Truth is Linear

Yining Chen·Jon A. Wellner
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Abstract

We prove that the convex least squares estimator (LSE) attains a $n^{-1/2}$ pointwise rate of convergence in any region where the truth is linear. In addition, the asymptotic distribution can be characterized by a modified invelope process. Analogous results hold when one uses the derivative of the convex LSE to perform derivative estimation. These asymptotic results facilitate a new consistent testing procedure on the linearity against a convex alternative. Moreover, we show that the convex LSE adapts to the optimal rate at the boundary points of the region where the truth is linear, up to a log-log factor. These conclusions are valid in the context of both density estimation and regression function estimation.

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Yining Chen, Jon A. Wellner. 2015-11-09. On Convex Least Squares Estimation when the Truth is Linear. https://doi.org/10.1214/15-ejs1098

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