On convex least squares estimation when the truth is linear

Electronic Journal of Statistics
Yining Chen, Jon A Wellner

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.

References

Sep 19, 2009·Scandinavian Journal of Statistics, Theory and Applications·Piet GroeneboomJon A Wellner
Nov 3, 2009·Annals of Statistics·Fadoua BalabdaouiJon A Wellner
Jul 1, 2009·Science in China. Series A, Mathematics·Gao Fuchang, Wellner Jon A
Apr 25, 2014·Bernoulli : Official Journal of the Bernoulli Society for Mathematical Statistics and Probability·Fadoua Balabdaoui, Jon A Wellner
Jan 1, 2016·Annals of Statistics·Charles R Doss, Jon A Wellner

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