On Convex Least Squares Estimation when the Truth is Linear

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.

35 pages, 5 figures

References in corpus (4)

• Asymptotics of the discrete log-concave maximum likelihood estimator and related applications
• Convergence of linear functionals of the Grenander estimator under misspecification
• Smoothed log-concave maximum likelihood estimation with applications
• Chernoff's density is log-concave

Cited by in corpus (1)

• Stratified incomplete local simplex tests for curvature of nonparametric multiple regression