Estimation of a convex function: characterizations and asymptotic theory.

By Piet Groeneboom, Geurt Jongbloed, Jon A. Wellner. 2001

179 citations recorded by the source corpus.

Subfields: Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/fa736936b515e4b72d1392405f466ae6c7ac75d4

Citations in the corpus

• Semiconvex Regression for Metamodeling-Based Optimization (2014, unknown): “Methods include the least squares estimator [39, 22, 34, 47, 29, 75, 24, 27], order restricted splines [71, 48, 49, 65], constrained kernel regression [11], and Bernstein polynomials [18].”
• Global risk bounds and adaptation in univariate convex regression (2013, unknown): “Mammen (1991) derived the rate of convergence of the LSE and its derivative at a fixed point, while Groeneboom et al. (2001) proved consistency and derived its asymptotic distribution at a fixed point of positive curvature.”
• Representation theorem for convex nonparametric least squares (2008, unknown): “Groeneboom et al. (2001) have derived the asymptotic distribution of the univariate CNLS estimator at a fixed point.”
• Generalized additive and index models with shape constraints (2014, unknown): “For the problem of univariate convex regression, see Hanson and Pledger (1976), Groeneboom et al. (2001, 2008) and Guntuboyina and Sen (2013).”
• Estimating a Convex Function in Nonparametric Regression (2007, unknown): “As pointed out by a referee it is of some interest to compare the new estimate m̂C proposed in this paper with the estimates obtained from the min-max counterpart of convex function estimation (see Groeneboom et al., 2001).”
• A general asymptotic scheme for inference under order restrictions (2006, unknown): “Define m̂ = T[0,1](xn(t)), and note in the following theorem that the rate n−2/5 in the independent data case is the same as the rate in the limit distribution result for the convex regression; see [23, 24].”
• Nonparametric Shape-Restricted Regression (2017, unknown): “The pointwise asymptotic distribution of the convex LSE (properly normalized) is derived in [61].”
• Global rates of convergence in log-concave density estimation (2014, unknown): “We handle the first difficulty with the aid of a recent result of Guntuboyina and Sen (2013), who showed that when working with the L2-metric (or more generally, Lr metrics with r ∈ [1,∞)), a metric entropy bound of the same order in ǫ can be obtained without the Lipschitz condition (but still with ”
• Estimation of a k-monotone density: limit distribution theory and the Spline connection (2005, unknown): “The arguments there are constructed along the lines of Groeneboom, Jongbloed and Wellner [18].”