Estimation of a convex function: Characterizations and asymptotic theory. (English) Zbl 1043.62027

Summary: We study nonparametric estimation of convex regression and density functions by methods of least squares (in the regression and density cases) and maximum likelihood (in the density estimation case). We provide characterizations of these estimators, prove that they are consistent and establish their asymptotic distributions at a fixed point of positive curvature of the functions estimated. The asymptotic distribution theory relies on the existence of an “invelope function” for integrated two-sided Brownian motion \(+t^4\) which is established in our companion paper ibid., 1620–1652, see the preceding entry, Zbl 1043.62025.


62G07 Density estimation
62G08 Nonparametric regression and quantile regression
62E20 Asymptotic distribution theory in statistics


Zbl 1043.62025
Full Text: DOI


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