Nonparametric estimation by convex programming

The problem we concentrate on is as follows: given (1) a convex compact set X in .n, an affine mapping x.A(x), a parametric family {p.(.)} of probability densities and (2) N i.i.d. observations of the random variable ., 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 {p.(.)} 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.