Nonparametric Estimation by Convex Programming

The problem we concentrate on is as follows: given (1) a convex compact set X in R, an affine mapping x 7→ 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 gx 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.