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 {pμ(-)} of probability densities and (2) N i.i.d. observations of the random variable ∈ X, 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 [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. © Institute of Mathematical Statistics, 2009.