Nonparametric estimation by convex programming
Anatoli B. Juditsky and Arkadi S. Nemirovski
Source: Ann. Statist. Volume 37, Number 5A
(2009), 2278-2300.
Abstract
The problem we concentrate on is as follows: given (1) a convex compact set X in ℝn, an affine mapping x↦A(x), a parametric family {pμ(⋅)} of probability densities and (2) N i.i.d. observations of the random variable ω, distributed with the density pA(x)(⋅) for some (unknown) x∈X, estimate the value gTx of a given linear form at x.
For several families {pμ(⋅)} with no additional assumptions on X and A, we develop computationally efficient estimation routines which are minimax optimal, within an absolute constant factor. We then apply these routines to recovering x itself in the Euclidean norm.
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Keywords: Estimation of linear functional; minimax estimation; oracle inequalities; convex optimization; PE tomography
Full-text: Open access
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Permanent link to this document: http://projecteuclid.org/euclid.aos/1247663755
Digital Object Identifier: doi:10.1214/08-AOS654
Zentralblatt MATH identifier: 05596901
Mathematical Reviews number (MathSciNet): MR2543692
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