Nonparametric estimation by convex programming

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 {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^Tx 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.