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Abstract: The problem we concentrate on is as follows: given 1 a convex compact set$X$ in ${\mathbb{R}}^n$, an affine mapping $x\mapsto Ax$, a parametric family$\{p {\mu}\cdot\}$ of probability densities and 2 $N$ i.i.d. observationsof the random variable $\omega$, distributed with the density $p {Ax}\cdot$for some unknown $x\in X$, estimate the value $g^Tx$ of a given linear format $x$. For several families $\{p {\mu}\cdot\}$ with no additionalassumptions on $X$ and $A$, we develop computationally efficient estimationroutines which are minimax optimal, within an absolute constant factor. We thenapply these routines to recovering $x$ itself in the Euclidean norm.



Author: Anatoli B. Juditsky, Arkadi S. Nemirovski

Source: https://arxiv.org/







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