The problem we concentrate on is as follows: given (1) a convex compact set X in R~n, an affine mapping x|→A(x), a parametric family {p_μ(_)} of probability densities and (2) N i.i.d. observations of the random variable ω, distributed with the density p_(A(x)) (_) for some (unknown) x ∈ X, estimate the value g~T x of a given linear form at x. For several families {p_μ(_)} 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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