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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

Type
Article accepté pour publication ou publié
External document link
http://arxiv.org/abs/math/0509080v1
Date
2010
Journal name
Statistica Neerlandica
Volume
64
Number
1
Publisher
Wiley
Pages
45-70
Publication identifier
http://dx.doi.org/10.1111/j.1467-9574.2009.00438.x
Metadata
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Author(s)
Wellner, Jon
Balabdaoui, Fadoua
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 / Keywords
shape constraints; completely monotone; least squares; maximum likelihood; minimax risk; mixture models; multiply monotone; non-parametric estimation; rates of convergence
JEL
C14 - Semiparametric and Nonparametric Methods: General

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