Nonparametric Estimation by Convex Programming
A. Juditsky and A. Nemirovski
Annals of Statistics 2008.

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; 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 gT x 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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