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.
Citation
Anatoli B. Juditsky. Arkadi S. Nemirovski. "Nonparametric estimation by convex programming." Ann. Statist. 37 (5A) 2278 - 2300, October 2009. https://doi.org/10.1214/08-AOS654
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