Estimation of a k-monotone density: characterizations, consistency and minimax lower bounds
Wellner, Jon; Balabdaoui, Fadoua (2010), Estimation of a k-monotone density: characterizations, consistency and minimax lower bounds, Statistica Neerlandica, 64, 1, p. 45-70. http://dx.doi.org/10.1111/j.1467-9574.2009.00438.x
TypeArticle accepté pour publication ou publié
External document linkhttp://arxiv.org/abs/math/0509080v1
Journal nameStatistica Neerlandica
MetadataShow full item record
Abstract (EN)The classes of monotone or convex (and necessarily monotone) densities on inline image can be viewed as special cases of the classes of k-monotone densities on inline image. These classes bridge the gap between the classes of monotone (1-monotone) and convex decreasing (2-monotone) densities for which asymptotic results are known, and the class of completely monotone (∞-monotone) densities on inline image. In this paper we consider non-parametric maximum likelihood and least squares estimators of a k-monotone density g0. We prove existence of the estimators and give characterizations. We also establish consistency properties, and show that the estimators are splines of degree k−1 with simple knots. We further provide asymptotic minimax risk lower bounds for estimating the derivatives inline image, at a fixed point x0 under the assumption that inline image.
Subjects / Keywordsshape constraints; completely monotone; least squares; maximum likelihood; minimax risk; mixture models; multiply monotone; non-parametric estimation; rates of convergence
Showing items related by title and author.